Harris's conjecture on Mantovan cohomology and supercuspidal representations

Let $(G,\mu)$ be a local Shimura datum as in the source, let $b\in\mathbf{B}(G,\mu)$, and let $M_S\subset M_b$ be a standard Levi subgroup. Let $\rho$ be a supercuspidal representation in $\operatorname{Groth}(M_S(\mathbb{Q}_p))$, let $I^{M_b}_{M_S}$ and $I^G_{M_S}$ denote normalized parabolic induction, let $LJ$ be the Jacquet–Langlands map, and let $\mathrm{Rel}^{G,\mu}_{M_S,b}$ be the set of pairs defined by the stated conjugacy and Kottwitz-invariant conditions. For each pair, let $r_{-\mu_S}$ be the corresponding representation of $\widehat{M_S}\rtimes W_{E_{\{\mu_S\}_{M_S}}}$. Harris's conjecture. The representations

$$\mathrm{Mant}_{G,b,\mu}\bigl(LJ(\delta^{\frac{1}{2}}_{G,P_b}I^{M_b}_{M_S}(\rho))\bigr)$$

and

$$[I^G_{M_S}(\rho)]\left[\bigoplus_{(M_S,\mu_S)\in\mathrm{Rel}^{G,\mu}_{M_S,b}}r_{-\mu_S}\circ LL(\rho)|_{W_{E_{\{\mu_S\}_{M_S}}}}|\cdot|^{-\langle\rho_G,\mu\rangle}\right]$$

are equal in $\operatorname{Groth}(G(\mathbb{Q}_p)\times W_{E_{\{\mu\}_G}})$. In particular, when $b$ is basic, the same equality holds with $LJ(I^G_{M_S}(\rho))$ on the left and the corresponding displayed expression on the right. This is a proposed restatement and slight generalization of Harris's conjecture; the source does not report a proof or resolution.