Finite groups with small centralizers of word-values

Given a positive integer m and a group-word w, we consider a finite group G such that w(G) ≠ 1 and all centralizers of non-trivial w-values have order at most m. We prove that if w=v(x1q1,⋯,xkqk), where v is a multilinear commutator word and q1, ⋯ , qk are p-powers for some prime p, then the order of G is bounded in terms of w and m only. Similar results hold when w is the nth Engel word or the word w= [xn, y1, ⋯ , yk] with k≥ 1