Convolution exhaustion conjecture for mirabolic character sheaves

Let $D$ be a vector space, and let $V$ and $W$ be vector spaces with $\dim(V)+\dim(W)=\dim(D)$. Let $\CG\in\calC(\operatorname{GL}(V))$ and $\CG'\in\calC(\operatorname{GL}(W))$, and let ${{\underline{\overline{\mathbb Q}}}{}_l}$ denote the unique equivariant mirabolic character sheaf for the zero-dimensional vector space. Convolution exhaustion conjecture. Any irreducible $\Gm$-equivariant mirabolic character sheaf on $\operatorname{GL}(D)\times D$ is isomorphic to

$$\CG*{{\underline{\overline{\mathbb Q}}}{}_l}*\CG'$$

for some such $\CG$ and $\CG'$. This asserts that all irreducible equivariant mirabolic character sheaves arise from the convolution construction; the supplied text gives no evidence that the conjecture has been resolved.