Generic transversality conjecture for opers with real monodromy

Let $\mathscr{X}$ be a sufficiently general pointed Riemann surface, and let $G$ be the reductive group occurring in the moduli spaces below. The spaces $\mathrm{Op}_P(\mathscr{X},G)$ and $\mathrm{Rep}_P^0(\Gamma,G(\mathbb{R}))$ are real submanifolds of $\mathrm{Rep}_P^0(\Gamma,G(\mathbb{C}))$, and write

$$\mathrm{Op}_P(\mathscr{X},G(\mathbb{R}))=\mathrm{Op}_P(\mathscr{X},G)\cap\mathrm{Rep}_P^0(\Gamma,G(\mathbb{R})).$$

Generic transversality conjecture. The two real submanifolds $\mathrm{Op}_P(\mathscr{X},G)$ and $\mathrm{Rep}_P^0(\Gamma,G(\mathbb{R}))$ intersect transversally at every point. Equivalently, $\mathrm{Op}_P(\mathscr{X},G(\mathbb{R}))$ is a discrete set.

The assertion predicts that the discreteness proved for permissible $G$-opers extends to all points for a sufficiently general pointed Riemann surface. It is presented as a general expectation in the source, and no resolution is supplied.