The $RC_k$-equivalence characterization for ribbon tubes

Let $k$ be a positive integer, and let $T,T'$ be ribbon tubes. They are $RC_k$-equivalent when they are related by the ribbon-tube analogue of surgery along trees of degree at least $k$, together with the corresponding equivalence moves. Two $RC_k$-equivalent ribbon tubes cannot be distinguished by finite type invariants of degree $<k$.

The $RC_k$-equivalence conjecture. Two ribbon tubes are $RC_k$-equivalent if and only if they cannot be distinguished by finite type invariants of degree $<k$.

The forward implication is known, and the converse holds for ribbon $2$-knots by Watanabe. The corresponding statement for ribbon tubes is presented as an open analogue of the welded-string-link conjecture.