Kottwitz conjecture for local shtuka cohomology

Let $G$ be a reductive group over $\bbQ_p$, let $b\in B(G)$ be basic, let $\mu_\bullet=(\mu_i)_{i\in I}$ be a finite collection of conjugacy classes of cocharacters, and let $\rho$ be an irreducible admissible representation of $G_b(\bbQ_p)$ with elliptic local Langlands parameter $\phi=\phi_\rho$. Thus $\phi$ is semisimple and

$$S_\phi/Z(\widehat{G})^{W_{\bbQ_p}}$$

is finite. Write $\operatorname{Irr}(S_\phi,\chi_b)\to\Pi_\phi(G)$ for the predicted surjection, $\delta\mapsto\pi_\delta$, and let $V_\phi$ be the associated $S_\phi\times W_{E_\bullet}$-representation. Define

$$\operatorname{Mant}_{G,b,\mu_\bullet}(\rho)=\sum_n(-1)^nH^n(R\Gamma(G,b,\mu_\bullet)[\rho]).$$

Kottwitz conjecture. One has

$$\operatorname{Mant}_{G,b,\mu_\bullet}(\rho)=\sum_{\delta\in\operatorname{Irr}(S_\phi,\chi_b)}\pi_\delta\boxtimes\operatorname{Hom}_{S_\phi}(\delta,V_\phi).$$

This is the predicted description of the intersection cohomology of local shtuka spaces in terms of the local Langlands correspondence; the accompanying text presents it as a natural generalization of the Kottwitz conjecture, but gives no resolution of the assertion here.