Torelli conjecture for polarized Hodge structures of opers

Let $\mathscr{X}$ be a pointed Riemann surface. For $i=1,2$, let $\mathscr{E}_{\odot,i}^{\spadesuit}$ be a $G^\odot$-oper classified by $\mathrm{Op}_P(\mathscr{X},G^\odot(\mathbb{R}))$, with monodromy map

$$\mu_i:=\mathrm{Mon}_{\mathscr{E}_{\odot,i}^{\spadesuit}}:\Gamma\longrightarrow G^\odot(\mathbb{R}).$$

Suppose that $\mu:=\mu_1=\mu_2$. For each positive integer $j$, the associated polarized real Hodge structures are defined on $H^1_P(\Gamma,V_{2j,\mathbb{R},\mu})$.

Torelli conjecture for opers. The following conditions are equivalent:

• $X_1$ is biholomorphic to $X_2$;
• the polarized real Hodge structures on $H^1_P(\Gamma,V_{2j,\mathbb{R},\mu})$ associated to $\mathscr{E}_{\odot,1}^{\spadesuit}$ and $\mathscr{E}_{\odot,2}^{\spadesuit}$ are identical, up to a constant factor, for every $j\in\mathbb{Z}_{>0}$.

Moreover, the equality of these data implies $\mathscr{E}_{\odot,1}^{\spadesuit}\cong\mathscr{E}_{\odot,2}^{\spadesuit}$, and each pair $(\mathscr{X},\mathscr{E}_{\odot}^{\spadesuit})$ with $\mathscr{E}_{\odot}^{\spadesuit}\in\operatorname{ob}(\mathrm{Op}_P(\mathscr{X},G^\odot(\mathbb{R})))$ can be reconstructed from its monodromy map and the associated polarized real Hodge structures.

This is a Torelli-type statement: it asks whether the pointed Riemann surface and the oper can be recovered from monodromy together with the Hodge-theoretic data. The source describes it as expected and does not provide a resolution.