Breuil-M\'ezard conjectures for central division algebras

We formulate an analogue of the Breuil-M\'ezard conjecture for the group of units of a central division algebra over a $p$-adic local field, and we prove that it follows from the conjecture for $\mathrm{GL}_n$. To do so we construct a transfer of inertial types and Serre weights between the maximal compact subgroups of these two groups, in terms of Deligne-Lusztig theory, and we prove its compatibility with mod $p$ reduction, via the inertial Jacquet-Langlands correspondence and certain explicit character formulas. We also prove analogous statements for $\ell$-adic coefficients.