The torus-torsor extension conjecture for $\Gamma_1(p)$-level covers

Assume the divisor conjecture and let $\pi:{\rm Sh}_{K_1}(G,X)_E \ifbool{@display}{\longrightarrow}{\rightarrow} {\rm Sh}_{K_0}(G,X)_E$ be the generic-fiber $T_{\mathcal G}(\mathbb F_p)$-torsor, represented by a pair $(Q_\pi,a_\pi)$. Let ${\mathcal S}_{K_0}$ be the integral model with local-model morphism $\varphi$. The torus-torsor extension conjecture. There \exists a $T_{\mathcal G}$-torsor $P$ over ${\mathcal S}_{K_0}$, an isomorphism $P[1/p]\simeq Q_\pi$, and an isomorphism of pairs

$$\bigl(L_*(P),L_*(\alpha^{-1})\circ a_\pi\bigr)\simeq\varphi^*\bigl({\rm P}^{(-1)}_{{\mathcal G},\mu},s_{{\mathcal G},\mu}\bigr).$$

This is the proposed torsorial construction underlying the global integral model at $\Gamma_1(p)$-level; it is stated conditionally on the divisor conjecture and is not resolved in the source.