Some, but not all chondrichthyans have labial cartilage,
(Fig. 1) those small, lateral, multi-part strips that bridge the gap between the upper and lower jaws (Fig. 2) and create shark cheeks (Fig. 3). There are no tetrapod homologs for labial cartilage. Megamouth, hammerhead and great white sharks don’t have labial cartilage. Neither do sawfish, skates or tiny Ozarcus.

Kilmpfinger and Kriwet 2020 composed a ‘brief historical review’:
“The origin and function of LCs (labial cartilages) have been discussed for almost 150 years.
Gegenbauer 1872: vestiges of pre-mandibular visceral arches,
Pollard 1895: remains of skeletal supports of a primitive set of oral cirrhi as in Amphioxus.
Swertzoff 1916: vestiges of two segments of visceral arches
Goodrich 1930: a secondary structure of no great morphological importance
Smith 1937: strengthen and mobilize the mouth corners.
Veran 1995: “without functional utility” and therefore “in decline”
Motta and Wilga 2001: very important to generate suction forces for ingesting prey items.”
Kilmpfinger and Kriwet 2020 continue:
“Phylogenetic signals or evolutionary pathways for the origin and distribution of LCs among chondrichthyan fishes also have been suggested. For example chimeroids, representing the sister group to all shark-like chondrichthyans and hybodontiforms, which forms the extinct sister group of Elasmobranchii (extant sharks, rays, skates), have five pairs of LCs (Maisey 1983; Didier 1995) that could be considered as the plesiomorphic condition for chondrichthyans and Klug (2010) hypothesized that two dorsal and two ventral pairs of LCs represent the plesiomorphic condition for modern sharks, which would indicate that LCs were reduced during the evolution of sharks.”
That phylogeny is wrong. It is essentially upside-down, according to the large reptile tree (LRT, 1839+ taxa, subset Fig. 1) where chimeroids and hybodontids are derived from other sharks.
Putting things in a phylogenetic context,
(Fig. 1) Chondrosteus, sturgeons and phylogenetically earlier taxa do not have labial cartilages. Tiny external LCs first appear in the flat, wide-mouth and toothless Early Silurian Loganiella. The extant whale shark (Rhincodon; Figs. 2, 3) is the closest living relative and it has internal LCs that create shark cheeks. The whale shark and manta ray demonstrate how Loganiella might have fed, but on a much larger scale (Fig. 3). The original deep open mouth of Chondrosteus turns into a deep closed tube with cheeks in Rhincodon (Fig. 3) due to internal labial cartilages. In the LRT labial cartilages appeared once and disappeared five times.
Imagine what your life would be like
without cheeks…
Basalmost chordates,
like Branchiostoma, Metaspriggina, Arandaspis, and Drepanaspis depend on deep cheeks to process plankton. Sturgeons have tubular mouth parts that extend ventrallly. When jaws first appear in Chondrosteus the tubes / cheeks go away, but quickly return with the appearance of external and internal labial cartilages in Loganiella, Manta and Rhincodon. Some mouths are more efficient with cheeks. Others are not.


When the mouth opens in basal taxa,
like Manta (Fig. 5) flexible external labial cartilages (= cephalic fins) form flexible side walls to funnel plankton-filled water into the shallow mouth and deep gill basket.

In taxa that attack prey larger than themselves,
like Daliatis (kitefin shark, Fig. 4) and Isistius (cookie cutter shark), labial cartilages can be relatively large. When the mouth is open and attached to prey, the labial cartilages and the cheeks they support keep the mouthful of ripped flesh inside the shark’s mouth.

Without a valid phylogenetic context,
and a last common ancestor, all hypotheses are likely to end up as wrong guesses, no matter how many µCT scans you use.
Klimpfinger and Kriwet conclude
“It is evident that position and dimension of LCs are connected to the shape of the jaws, forming distinct morphological modules.” That’s pretty generic, something that could have been stated prior to their study. Armed with a valid phylogenetic analysis the authors could have found a last common ancestor and traced the expansion, shrinkage and general evolution of labial cartilages among the various chondrichthyan clades (Fig. 1). Instead the authors, armed with a MSc and a PhD, left those tasks to an amateur blog poster working from data discovered on Google. Focused studies are great, but we still need to understand and confirm the systematics a little better. Taxon exclusion remains the number one problem in paleo.
References
Klimpfinger C and Kriwet J 2020. Comparative morphology of labial cartilages in sharks (Chondrichthyes, Elasmobranchii), The European Zoological Journal, 87:1, 741-753, DOI:
10.1080/24750263.2020.1844323