Scaffold exchange symmetry conjecture

Fix $n\ge1$, $k\ge0$, and $r\ge1$. Let $(a,w,S)\in\mathsf{LD}(n)^{\bullet k}$, let $m$ be a diagonal multiset of size $n$, and let $\Xi$ be a scaffold recording the diagonals, labels, and decoration indicators of rows with labels outside $\{r,r+1\}$. Define

$$\Phi^{(r)}_{m,k,\Xi}(u,v;q):=\sum_{(a,w,S)}q^{\operatorname{dinv}(a,w,S)}u^{\#\{j:w_j=r\}}v^{\#\{j:w_j=r+1\}}.$$

Scaffold exchange conjecture. For every $n\ge1$, $k\ge0$, $r\ge1$, every diagonal multiset $m$ of size $n$, and every scaffold $\Xi$,

$$\Phi^{(r)}_{m,k,\Xi}(u,v;q)=\Phi^{(r)}_{m,k,\Xi}(v,u;q).$$

This conjecture would establish the relevant color-exchange symmetry class by class and support the symmetry required by the valley Delta conjecture. It is presented as an open conjecture; the paper proves related low-area results.