On modular representations of inner forms of $\mathrm{GL}_n$ over a local non-archimedean field

Let $\mathrm{F}$ be a local non-archimedean field of residue characteristic $p$ and $\overline{\mathbb{F}}_\ell$ an algebraic closure of a finite field of characteristic $\ell \neq p$. We extend the results of Lapid and Mínguez concerning $\square$-irreducible representations of inner forms of $\mat