The sheet formulation of the covering-duality conjecture

Let $\fg$ be simple of simply-laced type, let $k\in\BQ$ satisfy $k+\check{\operatorname{\mathbb{h}}}=m/u$ in lowest terms with $m\geq1$ and $\operatorname{gcd}(m,u)=1$, and let $\check\BO(m)^{(u)}$ be the corresponding orbit in the metaplectic dual. Write $\check\BO(m)^{(u)}=\operatorname{Sat}_{\check L^{(u)}}^{\check G^{(u)}}\BO_{\check L^{(u)}}$, let $\fl$ be the corresponding Levi of $\fg$, let $\bd_L^{(u)}$ be the Levi covering-duality map, and let $\cS(\fl,\BO)$ denote the relevant sheet. Sheet formulation. One has

$$X_{L_k(\fg)}=\overline{\cS(\fl,\bd_L^{(u)}\BO_{\check L^{(u)}})}.$$

This is the sheet-theoretic restatement of the paper's main conjecture, since the corresponding sheet closure is identified with the induced expression in the main formulation. It should therefore be merged conceptually with that conjecture rather than treated as an independent claim.