The modular GKO coset construction conjecture for affine $\mathfrak{sl}_2$

Let ${\mathbb F}$ be an algebraically closed field of characteristic $p>2$, and let $\ell\geq 2$. For $0\leq n\leq \ell-1$ and $\epsilon\in\{0,1\}$, consider the multiplicity-free decomposition

$$V(\lambda_{\ell-1;n})\otimes V(\omega_\epsilon)=\bigoplus_{\substack{0\leq j\leq n\j\equiv n+\epsilon\pmod 2}}V(\lambda_{\ell;j})\otimes L_{c_\ell,h_{n+1,j+1}}\ \oplus\ \bigoplus_{\substack{n+1\leq j\leq \ell\j\equiv n+\epsilon\pmod 2}}V(\lambda_{\ell;j})\otimes L_{c_\ell,h_{\ell-n,\ell+1-j}}.$$

Here $V(\lambda_{\ell-1;n})$ and $V(\omega_\epsilon)$ are Weyl modules for affine $\mathfrak{sl}_2$ at levels $\ell-1$ and $1$, respectively, $V(\lambda_{\ell;j})$ is the Weyl module at level $\ell$, and $L_{c_\ell,h}$ is the irreducible Virasoro module of central charge $c_\ell$ and highest weight $h$.

Modular GKO conjecture. If $\operatorname{char}{\mathbb F}>2\ell^2+\ell-3$, then the multiplicity-free decomposition above is valid over ${\mathbb F}$ as a direct sum of irreducible $(\widehat{\mathfrak{sl}}_2|_\ell,\operatorname{Vir})$-modules.

This is the modular version of the Goddard--Kent--Olive coset construction, which is known over the complex numbers. Its validity in the stated characteristic range is conjectural and is tied to the expected irreducibility and rationality of the corresponding affine and Virasoro representation categories.