The canonical isomorphism conjecture for generalized Barsotti–Tate quotient 2-stacks

Let $G$ be a smooth affine group scheme over ${\mathbb{Z}}/p^n{\mathbb{Z}}$ and let $\mu:{\mathbb{G}}_m\to G$ be 1-bounded. Let $\operatorname{BT}_n^{G,\mu,?}$ be the quotient 2-stack obtained from the action of $G(\hat{\mathscr R}_n^\oplus)^{{\mathbb{G}}_m}$ on $G(\hat{\mathscr R}_n)$ by two-sided translations, and let $\operatorname{BT}_n^{G,\mu}$ be the 1-stack defined in GMM. Canonical isomorphism conjecture. There is a canonical isomorphism

$$\operatorname{BT}_n^{G,\mu}\buildrel{\sim}\over{\longrightarrow}\operatorname{BT}_n^{G,\mu,?}.$$

This conjecture identifies the quotient 2-stack arising from the explicit group-stack action with the established 1-stack of generalized truncated Barsotti–Tate groups, and would provide the desired explicit presentation in characteristic $p$.