  We deal with the following conjecture. If w is a group word and G 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 e and w 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 w is
a word and G 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).
