Whittaker multiplicity invariance under torus twisting

Let $\tilde{\varphi}_0:X_{Q,n}/X_{Q,n}^{sc}\to X/X_G^{sc}$ be well-defined, let $\mathfrak{w}=(B,U,\psi)$ be Whittaker data, and let $I(\overline{\overline{\overline{\overline{\chi}}}})$ be a $(K,s_K)$-unramified genuine principal series. Let \zeta_\overline{\overline{\overline{\overline{\chi}}}}:T_{ad}/T\twoheadrightarrow\operatorname{Irr}(\mathcal{S}(\phi_\overline{\overline{\overline{\overline{\chi}}}})) be the preceding map. Whittaker twisting conjecture. For every \pi(\phi_\overline{\overline{\overline{\overline{\chi}}}},\rho)\in\mathcal{L}(\phi_\overline{\overline{\overline{\overline{\chi}}}}) and every $t\in T_{ad}/T$,

\dim\operatorname{Wh}_\psi(\pi(\phi_\overline{\overline{\overline{\overline{\chi}}}},\rho))=\dim\operatorname{Wh}_{{}^t\psi}(\pi(\phi_\overline{\overline{\overline{\overline{\chi}}}},\zeta_\overline{\overline{\overline{\overline{\chi}}}}(t)\otimes\rho)).

This predicts invariance of Whittaker multiplicities under the corresponding torus action, with the packet member and Whittaker datum twisted simultaneously.