Lambertz et al. 2016 imagine
a diving aquatic niche for caseids like Cotylorhyhnchus (Fig. 1), and in order to breathe upon surfacing, a mammal-like diaphragm must have been present.
One of the authors, Dr. Steven Perry, has been working on the origin of the diaphragm for many years. Perry et al. 2010 wrote: “despite over 400 years of research into respiratory biology, the origin of this exclusively mammalian structure remains elusive.” (But see below)
According to Wikipedia: “Mammals have diaphragms, and other vertebrates such as amphibians and reptiles have diaphragm-like structures, but important details of the anatomy vary, such as the position of the lungs in the abdominal cavity.”
And “Tegu lizards are known to possess a proto-diaphragm, which separates the pulmonary cavity from the visceral cavity. While not actually capable of movement, it does allow for greater lung inflation, by taking the weight of the viscera off the lungs.”
And “Crocodilians have a muscular diaphragm that is analogous to the mammalian diaphragm. The difference is that the muscles for the crocodilian diaphragm pull the pubis (part of the pelvis, which is movable in crocodilians) back, which brings the liver down, thus freeing space for the lungs to expand.”
And this important and pertinent note to pet lizard owners:
“If you turn them over and stroke their bellies, they zonk out… Cute?.. NO, Stop! Lizards do not have diaphragms to help them breath. Their ribs moving in and out actually cause their lungs to inflate and deflate. When a dragon is held upside down or on its back, its stomach pushes on its lungs making it difficult for it to breath and will eventually result in suffocation.” Other similar cautionary notes are compiled here.
Unfortunately, Lambertz et al. also revert to an old invalid tradition,
that caseids are basal synapsids. For over five years it has been known that caseids are not basal to synapsids. The large reptile tree nests caseids as sisters to Feeserpeton and Australothyris and all are derived from a sister to Milleretta within the Lepidosauromorpha, not the Archosauromorpha, in which the Synapsida nests. Thus if you want to know if caseids had a diaphragm, you need to look at living lizards, all of which lack a working diaphragm.
Figure 1. Cotylorhynchus romeri. Extant lizards lack a diaphragm, so caseids also lacked a daphragm.
Given that backstory Lambertz et al. report:
“The origin of the diaphragm remains a poorly understood yet crucial step in the evolution of terrestrial vertebrates, as this unique structure serves as the main respiratory motor for mammals. Here, we analyze the paleobiology and the respiratory apparatus of one of the oldest lineages of mammal-like reptiles: the Caseidae.  Combining quantitative bone histology and functional morphological and physiological modeling approaches, we deduce a scenario in which an auxiliary ventilatory structure was present in these early synapsids. Crucial to this hypothesis are indications that at least the phylogenetically advanced caseids might not have been primarily terrestrial but rather were bound to a predominantly aquatic life. Such a lifestyle would have resulted in severe constraints on their ventilatory system, which consequently would have had to cope with diving-related problems.  Our modeling of breathing parameters revealed that these caseids were capable of only limited costal breathing and, if aquatic, must have employed some auxiliary ventilatory mechanism to quickly meet their oxygen demand upon surfacing.  Given caseids’ phylogenetic position at the base of Synapsida  and under this aquatic scenario, it would be most parsimonious to assume that a homologue of the mammalian diaphragm had already evolved about 50 Ma earlier than previously assumed.” 
- Not valid for the last five years. Caseids are derived from millerettids and are related to non-synapsids with a convergent lateral temporal fenestra. Hence the confusion.
- No one imagines caseids as divers. Maybe shoulder deep in shallow streams.
- Diving turtles have no such problems upon surfacing.
- Wrong again. See above.
- This is a ‘just-so’ story built on taxon exclusion and a couple of big IFs. See below for a hypothesis built on phylogenetic bracketing and skeletal morphology.
So while we’re on the topic of diaphragms,
let’s take a look at another possibility in stem mammals. Since basalmost mammals, like the platypus, Ornithorhynchus, have a diaphragm we’re looking for the origin of this lung muscle in earlier taxa.
A likely place to look
is at the transition from lateral undulation to limb rotation during locomotion. Only at that stage, where both lungs can inflate simultaneously during locomotion (see Carrier’s constraint), can the diaphragm develop.
Figure 2. Procynochus, Thrinaxoon, Chiniquodon transition to erect hind limbs while keeping sprawling forelimbs. This was the first stage in parasagittal locomotion, a requirement for the invention of the diaphragm and the most likely stage for its origin.
That transition began with the hind limbs on
derived cynodonts (Fig. 2) which slowly evolved parasagittally rotating hind limbs while retaining sprawling fore limbs. Monotreme mammals continue to retain sprawling forelimbs. Parasagittal forelimbs first appear with Juramaia and the later Therians.
The lumbar ribs began to shrink in derived cynodons (Fig. 2) disappearing completely in basalmost mammals.
The dorsal rib cage becomes pear-shaped in dorsal view (Fig. 3), with narrower ribs anteriorly and wider ribs posteriorly, near the developing diaphragm.
The dorsal vertebrae become differentiated into dorsal and lumbar vertebrae with neural spines angled posteriorly and anteriorly respectively and shorter and longer vertebral lengths respectively.
Sternal ribs, sternebrae, a manubrium and xiphoid process all appear in basalmost mammals, likely signaling the completion of the evolution of the diaphragm.
the vertebral column in vivo develop an arch in lateral view (Fig. 3) with a rise to the base of the rib cage followed by a lumbar decent to the sacrals.
The external nares become anteriorly oriented, confluent and the premaxillary ascending process disappears, facilitating greater volumes and velocities with every breath.
Figure 3 Megazostrodon, an a Jurassic mammal, along with Hadrocodium, a Jurassic tiny mammal.
in the transition from Cynodontia to Mammalia many changes occurred in the rib cage. Such changes are the most likely skeletal markers for the origin of the soft tissue diaphragm. Such changes are not seen in caseids, which, in any case, are related to lizards not mammals.
I have not read the Lambertz paper,
only the abstract, but with caseids unrelated to mammals, they are sadly barking up the wrong tree. Based on a false premise, that paper was a complete waste of time to produce. Build your papers on a solid phylogenetic foundation and everything will into place naturally.
Lambertz M, Shelton CD, Spindler F & Perry SF 2016. A caseian point for the evolution of a diaphragm homologue among the earliest synapsids. Annals of the New York Academy of Sciences (advance online publication) DOI: 10.1111/nyas.13264. http://onlinelibrary.wiley.com/doi/10.1111/nyas.13264/full
Merrell AJ and Kardon G 2013. Development of the diaphragm – a skeletal muscle essential for mammalian respiration. FEBS Journal 280(17): 4026-4035.
Perry SF, Similowski T, Klein W and Codd JR 2010. The evolutionary origin of the mammalian diaphragm. Repiratory Physiology & Nuerobiology 171(1):1-16.
Zimmer C. 2015. Behind Each Breath, an Underappreciated Muscle. The New York Times 04/07/2015.