The real-oper spectral support conjecture

Let $\operatorname{Oper}_{G^{\vee}}(\mathcal C)^{\mathbb R}$ denote the real locus of $G^{\vee}$-opers on a compact complex curve $\mathcal C$, and let $L^2(\operatorname{Bun})_o$ be the Hecke/differential-operator eigenspace indexed by an oper $o$. The real-oper spectral support conjecture.

$$L^2(\operatorname{Bun})_o\neq 0\quad\Longleftrightarrow\quad o\in\operatorname{Oper}_{G^{\vee}}(\mathcal C)^{\mathbb R}.$$

This identifies the spectral support with real opers and is part of the proposed multiplicity-one decomposition; the source gives no resolution status.