Quasi-modularity and MLDE conjecture for type-A generalized q-zeta values

Let $\mathfrak{g}=\mathfrak{sl}(n+1)$ with $n\geq 1$, let $k\geq 3$, and let $\mathcal{QM}=\mathbb{C}[E_2(\tau),E_4(\tau),E_6(\tau)]$ and $\mathcal{QM}[2]=\mathbb{Q}[E_2(\tau),\theta_2(\tau)^4,\theta_3(\tau)^4]$. Let ${\rm ch}[\mathbb{V}_{\mathfrak{g},k}](\tau)$ denote the character appearing in the vertex-algebra construction. Type-A quasi-modularity and MLDE conjecture. For even $k$, $\zeta_{\mathfrak{sl}(n+1),q}^k(k-2)\in\mathcal{QM}$, while for odd $k$ it belongs to $\mathcal{QM}[2]$; moreover, ${\rm ch}[\mathbb{V}_{\mathfrak{g},k}](\tau)$ is a solution of a modular linear differential equation with coefficients in $\mathcal{QM}$ or $\mathcal{QM}[2]$, according to the parity of $k$. These expectations are motivated by the $\mathfrak{sl}_2$ computations of Beem and Rastelli, who verified numerous examples; the general claim remains open.