Let S be a finite simple nonabelian group, w a word in a finite number of variables which is not a power of another word. Must there be a substitution of elements of S in w such that the resulting element is not 1?

Equivalently, let S be a finite simple group and F a free group on a finite number of variables. Let w be an element of F which is not a power of another element in F. Is there a homomorphism from F to S for which w is not in the kernel?