Subgroup membership problem in the braid group B4

Determine whether the subgroup membership problem is decidable in the four-strand braid group B4; specifically, decide whether there exists an algorithm that, given a finitely generated subgroup H ≤ B4 and an element g ∈ B4, determines whether g ∈ H.

Background

While Makanina proved undecidability of subgroup membership in Bn for n ≥ 5, the case n = 4 has resisted resolution. The authors note structural complexity (e.g., incoherence of B4) and provide various undecidability results, yet the subgroup membership problem for B4 remains unresolved.