Borel's pullback conjecture for enhanced local Langlands correspondences

Let $\tilde{d4a2}$ and $d4a2$ be connected reductive $F$-groups, and let $d45 d4a2d4a2d4a2$ be a homomorphism satisfying Condition 1: the kernel of its differential is central and its cokernel is a commutative $F$-group. Let $d4a2$ and $\tilde G$ be the corresponding groups of $F$-points, let $d4a2^L\eta$ be a dual L-homomorphism, and let ${}^S\eta:\mathbb C[\mathcal S_\phi]\to\mathbb C[\mathcal S_{{}^L\eta\circ\phi}]$ be the associated algebra homomorphism. For an enhanced parameter $(\phi,\rho)\in\Phi_e(G)$, write $\pi(\phi,\rho)$ for the corresponding representation.

Borel's pullback conjecture. Suppose that a local Langlands correspondence exists for sufficiently large classes of representations of $G$ and $\tilde G$. Then

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

This predicts the decomposition of pullback representations under a functorial local Langlands correspondence; the source gives no resolution status.