Twining big algebra conjecture for endoscopic groups

Let $\G$ be a connected semisimple complex Lie group with a distinguished automorphism $\sigma$ fixing a pinning, let $\G_\sigma=((\G^\vee)^\sigma_0)^\vee$ be the associated endoscopy group, and let $\mathcal B^\mu_\sigma(\mathfrak g)$ be the coinvariant algebra of the $\sigma$-action on the big algebra $\mathcal B^\mu(\mathfrak g)$. For $\mu\in\Lambda^+(\G_\sigma)$, also regard $\mu$ as the corresponding element of $\Lambda^+(\G)^\sigma$. Twining big algebra conjecture. There is an isomorphism

$$\mathcal B^\mu_\sigma(\mathfrak g)\cong\mathcal B^\mu(\mathfrak g_\sigma).$$

This identifies the coinvariant, or fixed-point, algebra associated with the automorphism $\sigma$ with the big algebra for the endoscopy Lie algebra $\mathfrak g_\sigma$. The source notes that a proof appeared in work of Zveryk, so the claim is solved rather than open.