Okounkov's structural conjecture for qMZVs

Let $\mathcal{Z}_q^{\mathrm{Oko}}$ be Okounkov's space of $q$-multiple zeta values, let $\operatorname{gr}^w_k$ denote its weight-graded pieces, and let $\zeta_q^{\mathrm{Oko}}(\mathbf{k})$ be the Okounkov $q$-multiple zeta values indexed by a tuple $\mathbf{k}$. Define $D(t):=\frac{1}{1-t^2}$, $O_3(t):=\frac{t^3}{1-t^2}$, and $S(t):=\frac{t^{12}}{(1-t^4)(1-t^6)}$.

Okounkov's conjecture. We have

$$\sum_{k\geq 0}\dim_{\mathbb{Q}}\left(\operatorname{gr}^w_k(\mathcal{Z}_q^{\mathrm{Oko}})\right)t^k=\frac{1}{1-t^2-t^3-t^4-t^5+t^8+t^9+t^{10}+t^{11}+t^{12}} =\frac{1}{(1-t^2)(1-t^4)(1-t^6)}\frac{1}{1-D(t)O_3(t)+2D(t)S(t)},$$

and $\mathcal{Z}_q^{\mathrm{Oko}}$ is spanned by the $\zeta_q^{\mathrm{Oko}}(\mathbf{k})$ with $2\leq k_i\leq 5$.

This conjecture predicts both the weight-graded dimensions of Okounkov's $q$-multiple zeta values and a finite-index spanning set. Its general validity remains open.