Descent of splendid Rickard equivalences in ${\mathrm{G}\mathrm{L}}_n(q)$

Let $n$ be a positive integer and $q$ a prime power. We prove that a refined version of Broué’s abelian defect group conjecture holds for unipotent $\mathrm{\ell }$-blocks of ${\mathrm{G}\mathrm{L}}_n(q)$, where $\mathrm{\ell }\nmid q$. We also give a sufficient condition on general $\mathrm{\ell }$-blocks of ${\mathrm{G}\mathrm{L}}_n(q)$ to satisfy the refined abelian defect group conjecture. We explain by an example that this sufficient condition does not hold in general.