Whittaker sheaf Hecke property conjecture

Let $G$ be the group and let $\gamma$ be the coweight used to define the stacks $\mathcal Y_d$. For a $\check{G}$-local system $E$ on $X$, let $\mathcal W^d_{E,\psi}$ be the Whittaker sheaf in the Grothendieck ring $K(\mathcal Y_d)$, and let $V^\gamma_E$ be the corresponding representation local system. Consider the Hecke correspondence

$$\mathcal Y_d\times X\xleftarrow{\mathfrak p_{\mathcal Y}\times\operatorname{supp}}\mathcal Y_d\times_{\operatorname{Bun}_G}\mathcal H_G^{+,1}\xrightarrow{\mathfrak q_{\mathcal Y}}\mathcal Y_{d+1}.$$

Whittaker sheaf Hecke property conjecture. There is a canonical isomorphism in $K(\mathcal Y_d\times X)$

$$(\mathfrak p_{\mathcal Y}\times\operatorname{supp})_!\mathfrak q_{\mathcal Y}^*\mathcal W^{d+1}_{E,\psi}\otimes\mathbb{\bar Q}_\ell(\tfrac12)[1]^{\otimes\langle\gamma,2\check{\rho}\rangle}\simeq\mathcal W^d_{E,\psi}\boxtimes V^\gamma_E\otimes\mathbb{\bar Q}_\ell(\tfrac12)[1].$$

This compatibility is intended to make the Whittaker sheaves into Hecke eigensheaves and is part of the proposed geometric construction. The source gives no resolution, so the conjecture remains open.