The schematized opposite-groupoid mirror conjecture

Let $Y$ be a compact Kähler manifold, let $Y_{top}$ denote its underlying topological space, and let $\mathcal{F}$ be a Kan-complex fibrant replacement of the singular simplicial set of $Y$, so that $\mathcal{F}$ models the fundamental $\infty$-groupoid $\Pi_{\infty}(Y)$. For a simplicial set or quasicategory $\mathcal{F}^{op}$, write $\mathcal{F}^{op}$ for its opposite $\infty$-groupoid, and let Toën's schematization functor be applied to its underlying simplicial set. Schematized opposite-groupoid mirror conjecture. The mirror of $Y$ can be taken to be the opposite $\infty$-groupoid $\mathcal{F}^{op}$, with a good approximation obtained by applying Toën's schematization functor to the simplicial set underlying $\mathcal{F}^{op}$. One may use $\Pi_{\infty}(Y)$ as a model for $\mathcal{F}$ and truncate to the corresponding $n$-groupoid according to the dimension. The proposed construction is intended to relate mirror symmetry to the Quillen–Segal formalism and descent for simplicial presheaves; the representability of the schematization's $\pi_0$ is left to be determined.