Ladder generalization of the cusp irreducibility criterion

Let ${\bf m}$ be a ladder multisegment, let ${\bf n}\in\operatorname{Mult}_k$, and let $\operatorname{LC}_k$, $\operatorname{RC}_k$, $\operatorname{LI}_k$, and $\operatorname{RI}_k$ denote the combinatorial and irreducibility conditions in Proposition~. For the one-segment multisegment $[a,a]$, that proposition identifies ${\rm LC}_k([a,a],{\bf n})$ with ${\rm LI}_k({\rm Z}([a,a]),{\rm Z}({\bf n}))$ and ${\rm RC}_k([a,a],{\bf n})$ with ${\rm RI}_k({\rm Z}([a,a]),{\rm Z}({\bf n}))$.

Ladder generalization conjecture. The same equivalences remain true when $[a,a]$ is replaced by any ladder ${\bf m}$.

This would extend the known criterion from a single segment to arbitrary ladders. The source gives no resolution status.