Decidability of the mixed quantifier fragment for powers of two and three

Determine whether the \(\exists^*\forall^*\) fragment of the first-order theory of \(\langle\mathbb{N};0,1,<,+,\{2^n:n\in\mathbb{N}\},\{3^n:n\in\mathbb{N}\}\rangle\) is decidable.

Background

The paper discusses the structure obtained by expanding Presburger arithmetic with the predicates of powers of two and powers of three. The full first-order theory is known to be undecidable, while the existential fragment is known to be decidable. The cited results leave the intermediate $\exists^*\forall^*$ fragment unresolved.