Okounkov's dimension conjecture for q-multiple zeta values

Let ${\mathbf Z}_w^{\rm O}$ be the $\Q$-vector space generated by Okounkov's type O $q$-multiple zeta values $\frakz_q^{\rm O}[\bfs]$ with $|\bfs|\leq w$. Okounkov's dimension conjecture.

$$\sum_{w=0}^\infty t^w \dim {\mathbf Z}_{\leq w}^{\rm O}=\frac{1}{1-t-t^2+t^6+t^8-t^{13}}-\frac{1}{1-t}$$

and hence

$$\sum_{w=0}^\infty t^w \dim {\mathbf Z}_{\leq w}^{\rm O}=t^2+2t^3+4t^4+7t^5+11t^6+18t^7+27t^8+42t^9+63t^{10}+95t^{11}+142t^{12}+O(t^{13}).$$

The formula is rigorously verified for weights at most $6$ and numerically for weights at most $12$ in the source; its general validity remains open.