Irreducibility conjecture for geometric Eisenstein series

Let $G$ be a reductive group, let $T$ be a maximal torus, and let ${\mathcal S}\in \Sh(\Bun_T)^{reg}\cap \Perv(\Bun_T)$. The object $\Eis({\mathcal S})$ is a perverse sheaf.

Irreducibility conjecture. Moreover, if ${\mathcal S}$ is irreducible, then so is $\Eis({\mathcal S})$.

This asserts that geometric Eisenstein series preserve perversity and irreducibility for regular perverse sheaves on the torus-moduli stack. The supplied text gives no resolution status.