The chiral D-affineness conjecture for quotient stacks

Let $G$ be a semisimple group, let $R$ be a representation of $G$, and let $t$ lie in the Lie algebra of the compact form of $G$. Suppose that

$$\operatorname{Tr}_R(t^2)>\operatorname{Tr}_{\mathfrak{g}}(t^2).$$

Let $V$ be any representation of $G$ satisfying

$$\operatorname{Tr}_R(t^2)-\operatorname{Tr}_{\mathfrak{g}}(t^2)=\operatorname{Tr}_V(t^2).$$

Chiral D-affineness conjecture. The stack quotient $(R\oplus\Pi V)/G$ is chiral $D$-affine.

This would justify identifying global modules for the relevant chiral differential-operator vertex algebra with modules over its sheaf-theoretic counterpart on the Higgs branch supplemented by fermions. The source presents this as a conjectural condition needed for its global construction and gives no resolution.