Symmetric quasi-modularity conjecture for type-A bi-brackets

Let $n\geq 1$, let $k_{[i,j]}\geq 1$ and $s_{[i,j]}\geq 0$ be integers for $1\leq i\leq j\leq n$, and let $\zeta_{\mathfrak{sl}(n+1),q}[\cdots]$ denote the associated type-A bi-bracket series. Let $S_{n(n+1)/2}$ act by permuting the indexed entries. Type-A bi-bracket symmetry conjecture. If all $k_{[i,j]}$ are even, then

$$\sum_{\sigma\in S_{n(n+1)/2}}\zeta_{\mathfrak{sl}(n+1),q}^{k+2}(k_{\sigma[1,1]},\ldots,k_{\sigma[n,n]})\in\mathcal{QM}.$$

This is motivated by numerical experiments and by the bi-bracket construction for ordinary multiple $q$-zeta values; no general proof or resolution is given.