The Cox ring surjectivity conjecture for quiver flag varieties

Let $\mathcal{M}_\vartheta$ be a quiver flag variety, let $\lvert\underline{\det}(\mathscr{W})\rvert$ be the toric variety associated with the sequence of determinant line bundles, and let $Q'$ be its quiver. For each $\theta\in\mathbb{Z}^\rho$, let

$$g_\theta\colon H^0\big((\mathscr{W}'_1)^{\theta_1}\otimes\dots\otimes(\mathscr{W}'_\rho)^{\theta_\rho}\big)\longrightarrow H^0\big(\det(\mathscr{W}_1)^{\theta_1}\otimes\dots\otimes\det(\mathscr{W}_\rho)^{\theta_\rho}\big)$$

be the map induced by the multigraded Plücker morphism. Cox ring surjectivity conjecture. The homomorphism of $\mathbb{Z}^\rho$-graded $\Bbbk$-algebras

$$\bigoplus_{\theta\in\mathbb{Z}^\rho}g_\theta\colon \Bbbk[y_a:a\in Q'_1]\longrightarrow \operatorname{Cox}(\mathcal{M}_\vartheta)$$

is surjective. The maps are known to be surjective in the standard basis degrees, while the conjecture asserts surjectivity in all multidegrees; the source gives no resolution status.