Are compact open-charm tetraquarks consistent with recent lattice results?

We argue that the hypothesis that positive-parity charm meson resonances exhibit a compact tetraquark structure has some clear tension with recent lattice results for the $S$-wave $\pi D$ system for an SU(3) flavor symmetric setting. In particular, we show that such a diquark--anti-diquark tetraquark scenario would call for the presence of a state in the flavor $[{\mathbf{\overline{15}}}]$ representation, not seen in the lattice analysis. Moreover, we show that analogous lattice data in the axial-vector channel are even more sensitive to the internal structure of these very interesting states.