The length of mixed identities for finite groups

We prove that there exists a constant $c>0$ such that any finite group having no non-trivial mixed identity of length $\leq c$ is an almost simple group with a simple group of Lie type as its socle. Starting the study of mixed identities for almost simple groups, we obtain results for groups with socle ${\rm PSL}_n(q)$, ${\rm PSp}_{2m}(q)$, ${\rm P \Omega}_{2m-1}^\circ(q)$, and ${\rm PSU}_n(q)$ for a prime power $q$. For such groups, we will prove rank-independent bounds for the length of a shortest non-trivial mixed identity, depending only on the field size $q$.