Okounkov's derivation conjecture for q-multiple zeta values

Let $\operatorname{\mathsf{qMZV}}$ denote the algebra of $q$-multiple zeta values, containing the ring of quasi-modular forms $\widetilde{M}_{\mathbb{Q}}=\mathbb{Q}[G_2,G_4,G_6]$, and let $\operatorname{d}:=q\frac{d}{dq}$ be the $q$-derivative operator. Okounkov's conjecture. The operator $\operatorname{d}$ is a derivation on $\operatorname{\mathsf{qMZV}}$. This conjecture extends the known fact that the ring of quasi-modular forms is closed under the $q$-derivative; the cited result establishes that $\operatorname{d}$ is a derivation on the larger algebra $\operatorname{\mathcal{MD}}$, but it remains open whether it preserves $\operatorname{\mathsf{qMZV}}$.