Conjecture on the subregular unipotent Arthur sheaf for G2

Let $k$ be the ground field, let $G$ be the reductive group in the paper, and let $X$ be the underlying curve. Let $\Bun^d_{B,\Omega}$ denote the connected component of the relevant moduli stack with degree

$$d=\dim H^0\left(X,\frac{U_{\mathfrak b}}{T}\right).$$

The subregular Arthur-sheaf conjecture. If $\operatorname{char}(k)\neq 2$ or $3$, then there exists a unique sheaf $\mathcal F_{\mathrm{sr}}$ on $\operatorname{Bun}_G$ such that, on every connected component $\Bun^d_{B,\Omega}$ with $d$ sufficiently large, one has, up to a shift,

$$(\pi_P)^*\mathcal F_{\mathrm{sr}}\xrightarrow{\sim}(\pi_{\mathcal Y})_!j_{!*}(\pi_B)^*\mathcal F_{\mathrm{blue}}(\pi_!\overline{\mathbb Q}_\ell).$$

This sheaf satisfies the Hecke property for the subregular unipotent Arthur parameter $\sigma:\operatorname{SL}_2\to\check{G}$. The claim is a conjectural construction of the automorphic sheaf attached to the subregular unipotent parameter; the supplied text gives no resolution.