Group problem in the braid group B4

Determine whether the group problem is decidable in the four-strand braid group B4; that is, decide whether there exists an algorithm that, given a finite set X of braids, determines whether Sgp(X) is a subgroup of B4.

Background

For B3, the rational subset membership problem is decidable, implying decidability of several related problems, whereas for Bn with n ≥ 5, many problems (including the group problem) are undecidable via embeddings of F2 × F2. The status for B4 is delicate: the paper proves multiple undecidability results but leaves the group problem unresolved.