The global integral-model conjecture for $\Gamma_1(p)$-level Shimura varieties

Assume that $({\mathcal G},\{\mu\})$ is strictly convex, that the divisor conjecture holds, and that an $O_E$-integral model ${\mathcal S}_{K_0}$ of ${\rm Sh}_{K_0}(G,X)_E$ with its local-model morphism $\varphi$ is given. Let $S\subset S_{{\mathcal G},\mu}$ be a semigroup defining $Y_S$ and its Lang cover. The global $\Gamma_1(p)$-integral-model conjecture. There exists an $O_E$-integral model ${\mathcal S}_{K_1,S}$ of ${\rm Sh}_{K_1}(G,X)_E$ equipped with a morphism to ${\mathcal S}_{K_0}$ extending the generic-fiber cover, an extended $T_{\mathcal G}(\mathbb F_p)$-action identifying ${\mathcal S}_{K_0}$ with the quotient, the stated prime-to-$p$ equivariance, and a morphism to $[\mathcal G\backslash{\rm M}^{\sqrt S}_{{\mathcal G},\mu}]$ inducing the specified $2$-cartesian quotient-stack isomorphism. This is the proposed systematic construction of integral models at $\Gamma_1(p)$-type level; the source presents it as conjectural and does not claim a general proof.