The local model conjecture for tubular neighborhoods of p-adic shtuka spaces

Let ${{\mathcal{M}}^{\mathscr{G},\leq \mu}_{O_E}}$ be the local model and let ${{\mathcal{A}}_{\mathscr{G},\mu}}=({{\mathcal{M}}^{\mathscr{G},\leq \mu}_{O_E}})^{\mathrm{red}}$ be its reduced special fiber. Let $F\supseteq \breve{E}$ be a nonarchimedean field extension with ring of integers $O_F$ and algebraically closed residue field $k_F$. For a closed point $x\in |({X_{\mathscr{G}}^{\leq\mu}(b)})_{{k_F}}|$, write $({{\mathrm{Sht}}^{\mathscr{G}_b,\leq \mu}_{{O_F}}})_x^{\odot}$ for its formal neighborhood, and similarly write $({{\mathcal{M}}^{\mathscr{G},\leq \mu}_{O_F}})_y^{\odot}$ for the formal neighborhood of a closed point $y$.

Local model conjecture. For every closed point $x\in |({X_{\mathscr{G}}^{\leq\mu}(b)})_{{k_F}}|$ there exists a closed point $y\in |({{\mathcal{A}}_{\mathscr{G},\mu}})_{k_F}|$ such that the formal neighborhoods

$$({{\mathrm{Sht}}^{\mathscr{G}_b,\leq \mu}_{{O_F}}})_x^{\odot}\quad\text{and}\quad({{\mathcal{M}}^{\mathscr{G},\leq \mu}_{O_F}})_y^{\odot}$$

are isomorphic v-sheaves.

This conjecture is philosophically aligned with Grothendieck--Messing theory and predicts that tubular neighborhoods of points in the reduced special fiber of the shtuka space are modeled by formal neighborhoods in the local model. The surrounding results establish specialization and connected-component statements, while this local comparison is formulated as a conjectural final ingredient.