Goussarov--Habiro conjecture for string links in the cylinder

Let $L(m)$ denote the string links on $m$ strands. For $n\geq 1$, let $V_n$-equivalence be the equivalence relation induced by the Vassiliev filtration, and let $C_n$-equivalence be the equivalence relation generated by $C_n$-moves, namely clasper surgeries modeled on tree claspers of degree $n$. Goussarov--Habiro conjecture. For all $m\geq 1$ and $n\geq 1$, any $V_n$-equivalent string links on $m$ strands are $C_n$-equivalent. Goussarov and Habiro proved the converse implication for all string links, and proved this conjectured implication for knots, the case $m=1$; the assertion for general string links remains open.