Bounding the exponent of a verbal subgroup

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 \(n\text{ th }\) Engel word or the word \(x^n,y_1,y_2,\ldots ,y_k]\) (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).