Conjectural invariant six-dimensional subvariety of the special fiber

Let $\mathcal R$ be sufficiently small, and let $\mathfrak M_{\mathcal R,\Omega}$ be the special moduli space over $\mathbb F_{p^r}$ equipped with its $Z(\mathbb A^{\infty,p})\times G(\mathbb A^{\infty,p}\otimes L^+)$-action. Let $Y^{(1)}[\mathfrak q^\infty]$ denote the relevant formal $\mathfrak q$-divisible group.

Invariant-subvariety conjecture. The special fiber of $\mathfrak M_{\mathcal R,\Omega}$ contains a nonempty $Z(\mathbb A^{\infty,p})\times G(\mathbb A^{\infty,p}\otimes L^+)$-invariant subvariety $\overline{\mathfrak M}$ of dimension $6$ over $\mathbb F_{p^r}$ such that the formal isogeny types of $Y^{(1)}[\mathfrak q^\infty]$ at its points with values in a perfect field are among

$$G_{0,1}^{2r}\oplus G_{1,r-1}^3\oplus G_{2,r-2}^2,$$ $$G_{0,1}^r\oplus G_{1,2r-1}\oplus G_{1,r-1}\oplus G_{3,2r-3}\oplus G_{2,r-2},$$ $$G_{1,2r-1}\oplus G_{1,r-1}^3\oplus G_{3,2r-3},$$

and

$$G_{1,r-1}^7.$$

These give locally closed subvarieties of dimensions $6$, $5$, $4$ and $3$, respectively. Moreover, $G(\mathbb A^{\infty,p}\otimes L^+)$ acts trivially on the set of irreducible components of $\overline{\mathfrak M}\times_{\mathbb F_{p^r}}\overline{\mathbb F}_p$.

The conjecture seeks a smaller and more canonical variety inside the special fiber, with controlled formal isogeny strata and trivial action on irreducible components. The supplied text gives no resolution status.