Strongly productive ultrafilters on semigroups

We prove that if S is a commutative semigroup with well-founded universal semilattice or a solvable inverse semigroup with well-founded semilattice of idempotents, then every strongly productive ultrafilter on S is idempotent. Moreover we show that any very strongly productive ultrafilter on the free semigroup with countably many generators is sparse, answering a question of Hindman and Legette Jones.