Quasi-modularity of symmetrized generalized multiple q-zeta values

Let $\Delta$ be a root system of ADE type, let $\Delta_+$ be its set of positive roots, and let $r=|\Delta_+|$. For integers $k_\alpha\geq 1$, write

$$\zeta_{\mathfrak{g},q}(k_1,\ldots,k_r)$$

for the associated generalized multiple $q$-zeta value, and let $\mathcal{QM}=\mathbb{C}[E_2(q),E_4(q),E_6(q)]$ be the ring of quasi-modular forms on $SL(2,\mathbb{Z})$. The generalized q-zeta quasi-modularity conjecture. If $k_\alpha=2k$ for every positive root, then

$$\sum_{\sigma\in S_r}\zeta_{\mathfrak{g},q}(2k_{\sigma(1)},\ldots,2k_{\sigma(r)})\in\mathcal{QM};$$

in particular, $\zeta_{\mathfrak{g},q}(2k,\ldots,2k)\in\mathcal{QM}$. This extends the familiar $\mathfrak{sl}_2$ quasi-modularity phenomenon for even $q$-zeta values; the conjecture is presented as an expected symmetry formula, and no resolution is given.