Model companions and the Burnside problem for groups of fixed exponent

Determine whether, for every integer n>1, the theory T_n of groups of exponent n has a model companion if and only if every finitely generated group of exponent n is finite, equivalently, if and only if the Burnside problem has a positive solution for exponent n.

Background

For each integer n>1, T_n denotes the theory of groups satisfying the identity x^n=1, while the Burnside problem asks whether every finitely generated group of exponent n is finite. The paper studies the relationship between this local-finiteness property and the existence of a model companion for T_n.

The paper proves the n=3 case of the proposed equivalence and notes that the conjecture also holds for n=2 and for all sufficiently large primes, but leaves the general statement unresolved. Thus, the open problem is to establish or refute the equivalence for all remaining exponents.