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 $\le c$ 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 ${\mathrm{P}\mathrm{S}\mathrm{L}}_n(q)$, ${\mathrm{P}\mathrm{S}\mathrm{p}}_{2m}(q)$, ${\mathrm{P}\mathrm{\Omega }}_{2m-1}^{\circ }(q)$, and ${\mathrm{P}\mathrm{S}\mathrm{U}}_n(q)$ 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 $q$.