Precise pullback conjecture for local Langlands correspondences and Hecke algebras

Let $f:\tilde{\mathcal G}\to\mathcal G$ satisfy Condition 1, and use the corresponding notion of relevance for enhanced L-parameters. Let $\tilde\phi={}^Lf(\phi)$, and denote by $\pi(\phi,\rho)$ the representation associated to $(\phi,\rho)$.

Precise pullback conjecture. For every $(\phi,\rho)\in\Phi_e(G)$,

$$f^*(\pi(\phi,\rho))=\bigoplus_{\tilde\rho\in\operatorname{Irr}(\mathcal S_{\tilde\phi})}\operatorname{Hom}_{\mathcal S_\phi}\bigl(\rho,{}^Sf^*(\tilde\rho)\bigr)\otimes\pi(\tilde\phi,\tilde\rho).$$

This is a more precise version of the earlier local pullback conjecture and is intended to describe compatibility between local Langlands correspondences and pullback through the associated Hecke-algebra parametrizations. The source gives no resolution status.