Haines's twisted endoscopic transfer conjecture for the stable Bernstein center

Let $G_r$ be an unramified base-change form of the quasi-split group $G$, with Hecke algebras $\mathcal{H}(G_r)$ and $\mathcal{H}(G)$, and let $\mathfrak{Z}^{\textnormal{st}}(G_r)$ and $\mathfrak{Z}^{\textnormal{st}}(G)$ denote their stable Bernstein centers. Let

$$b_r':\mathfrak{Z}^{\textnormal{st}}(G_r)\rightarrow \mathfrak{Z}^{\textnormal{st}}(G)$$

be the base change homomorphism, viewed through the natural embedding of the stable Bernstein center into the Bernstein center, and assume that $G$ satisfies LLC+. Two functions are associated in the sense of base change if they have matching orbital integrals. Haines's conjecture. If $\phi\in\mathcal{H}(G_r)$ and $f\in\mathcal{H}(G)$ are associated, then $Z_r*\phi$ and $b_r'(Z_r)*\phi$ are associated for any $Z_r\in\mathfrak{Z}^{\textnormal{st}}(G_r)$. This conjecture generalizes the base change fundamental lemmas for spherical and parahoric Hecke algebras and for the Bernstein center of the depth-zero principal series block. Its status is not resolved in the supplied source context.