Identity problem in the braid group B4

Determine whether the fixed-target subsemigroup membership problem for the target element 1 (the identity problem) is decidable in the four-strand braid group B4.

Background

The authors prove several undecidability results in B4, including undecidability of a fixed-target submonoid membership problem for a specific nontrivial element and undecidability of the subsemigroup intersection problem. Despite these, the decidability of the identity problem (target 1) in B4 is not settled. This stands in contrast with B3, where many related problems are decidable, and Bn for n ≥ 5, where strong undecidability phenomena are known via embeddings of F2 × F2.