The anabelian restricted Burnside problem

Let $n,d \in \mathbb N$ and $w \in \mathbb F_n$ be non-trivial. We prove that the relatively free group of rank $d$ in the variety defined by the group law $w$ has a largest anabelian finite quotient and estimate its size. Here, a finite group is called anabelian if it has only non-abelian composition factors. The estimate is based on explicit bounds for the length of laws for finite simple groups obtained by Bradford and the author and on recent work by Fumagalli--Leinen--Puglisi.