GB-problem in the class of locally graded groups

A problem, we consider, is equivalent to the one posed in 1981 by Bergman: Let G be a group and S a subsemigroup of G which generates G as a group. Must each identity satisfied in S be satisfied in G? The first counterexample was found in 2005 by Ivanov and Storozhev. It gives a negative answer to the problem in general. However we show that the problem has an affirmative answer for locally residually finite groups and for locally graded groups containing no free noncyclic subsemigroups.