The $w_k$-equivalence characterization for welded string links

Let $k$ be a positive integer, and let $L,L'$ be welded string links. They are $w_k$-equivalent if one can be obtained from the other by a finite sequence of surgeries along $w_l$-trees with $l\geq k$, together with generalized Reidemeister and OC moves. Finite type invariants of degree $<k$ are invariants of $w_k$-equivalence.

The $w_k$-equivalence conjecture. Two welded string links are $w_k$-equivalent if and only if they cannot be distinguished by finite type invariants of degree $<k$.

The converse is known for welded knots and welded long knots, but remains open for welded string links. This would characterize precisely the information detected by finite type invariants of welded string links.