…not an armadillo.
The pink daily ‘armadillo’ (genus: Chlamyphorus) nests with the much larger glyptodont, Holmesina in the large reptile tree (LRT, 1255 taxa, subset Fig. 4), not with Dasypus novemcinctus, the extant llong-nosed armadillo nesting on the other side of the aardvark Orycteropus. Another fairy armadillo (genus: Calyptophractos) is also described here.
The pink fairy armadillos
(genus: Chlamyphorus trunca) and the greater fairy armadillo (genus: Calyptophractos retusus) are little glyptodonts. This is not heretical news (Fig. 1).

Figure 1. When glyptodonts were nested with armadillos, the fairy armadillos nested with extinct glyptodonts. Cladogram from Delsuc et al. 2016, a DNA analysis. No aardvarks were tested here.
Using DNA
(both ancient and modern) Delsuc et al. 2016 nested the fairy armadillo with the extinct glyptodont, Doedicurus (Fig. 1). We already know not to trust DNA evidence in paleontology, but in this case trait analysis supports something like this arrangement of taxa. The distance is not great either way.

Figure 2. DNA analysis by Möller-Krull et al. 2007 omits fossil taxa and aardvarks and arrives at this tree topology of extant edentates.
Delsuc et al. 2016 nested fairy armadillos with glyptodonts
when they attempted to nest glyptodonts within the armadillo clade using DNA, omitting other fossil taxa. I did not see LRT outgroups in their cladogram, nor did I see the aardvark.

Figure 3. Skulls of Holmesina and Calyptophractus compared. When they are together, the similarities are obvious.
Here in the LRT Calyptophractus is a phylogenetic miniature of Holmesina, with a shorter rostrum and expanded cranium (Fig. 3), along with its much smaller size and thinner scales. In lateral view the skulls are quite alike and distinct from all other edentates.
We know that aardvarks (genus: Orycteropus) nest with edentates because all the other possibilities were offered and found to be not as parsimonious (similar). Earlier we looked at the nesting of Holmesina and the phylogenetic fact that all aardvarks, armadillos and anteaters are derived from various types of glyptodonts.

Figure 4. Subset of the LRT focusing on the Edentata. Armored taxa are color tinted and their branches are thicker.
References
Delsuc F et al. 2016. The phylogenetic affinities of the extinct glyptodonts. Current Biology 26(4):R155–R156.
Harlan R 1825. Annals of the Lyceum of Natural History of New York 1:235.
Möller-Krull M et al. 2007. Retroposed Elements and Their Flanking Regions Resolve the Evolutionary History of Xenarthran Mammals (Armadillos, Anteaters, and Sloths). Mol. Biol. Evol. 24 (11): 2573–2582. PDF
Yarrell W 1828. On the osteology of the Chlamyphorus truncatus of Dr. Harlan microform; in a letter to N.A. Vigors. Zoological Journal 3:544–554.
AKA
Pink Fairy Armadillo, Lesser Fairy Armadillo, Lesser Pichi Ciego, Pichiciego
http://www.iucnredlist.org/details/4704/0
Greater Fairy Armadillo, Burmeister’s Armadillo, Chacoan Fairy Armadillo, Greater Pichi Ciego: http://www.iucnredlist.org/details/4703/0