Poset-type conjecture for quadratic multiplicities of key-polynomial generating series

Let $w$ be a permutation, let $k$ and $l$ be positive integers, and let $A_l(w)$ and $A_k(w)$ be the sets of bounded increasing sequences associated with $w$. For a multi-set $\eta$, write $m^{k,l}_{\eta}(w)$ for its multiplicity, and partially order the set of presentations $\alpha+\beta=\eta$ with $\alpha\in A_l(w)$ and $\beta\in A_k(w)$ as described in the paper.

Poset-type conjecture. The multiplicity $m^{k,l}_{\eta}(w)$ depends only on the poset type of the set of sums $\alpha+\beta=\eta$, with $\alpha\in A_l(w)$ and $\beta\in A_k(w)$. Moreover, when one poset can be embedded into another, the associated multiplicity either remains the same or increases.

This extends the results known for $r\leq2$ to all values of $r$ and predicts that the quadratic multiplicities are governed solely by the combinatorial type of their presentation posets. The conjecture is posed after checking additional cases; no resolution is supplied in the source.