Bounding the exponent of a verbal subgroup

We deal with the following conjecture. If is a group word and is a finite group in which any nilpotent subgroup generated by w-values has exponent dividing e, then the exponent of the verbal subgroup w(G) is bounded in terms of and only. We show that this is true in the case where w is either the nth Engel word or the word [x(n), y(1), y(2), ..., y(k)] (Theorem A). Further, we show that for any positive integer e there exists a number k = k(e) such that if is a word and is a finite group in which any nilpotent subgroup generated by products of k values of the word w has exponent dividing e, then the exponent of the verbal subgroup w(G) is bounded in terms of e and w only (Theorem B).