Kerz's commutativity conjecture for isomonodromic Higgs-bundle deformations

Fix $0\in S$, let $\mathbb V$ be an irreducible $\mathbb C$-local system on $X_0$, and let $\sigma_{\operatorname{Dol}}:S\to M_{\operatorname{Dol}}(X/S)$ be the real analytic isomonodromic deformation obtained from its de Rham isomonodromic deformation by the relative non-abelian Hodge correspondence. Writing $\sigma_{\operatorname{Dol}}(s)=[(E_s,\theta_s)]$, define

$$(\lambda\cdot\sigma_{\operatorname{Dol}})(s)=[(E_s,\lambda\cdot\theta_s)].$$

Here $S^1=\{z\in\mathbb C^*:|z|=1\}$.

Kerz's commutativity conjecture. (i) For any $\lambda\in\mathbb R^*$, $\lambda\cdot\sigma_{\operatorname{Dol}}$ is also an isomonodromic deformation. (ii) For any $\lambda\in S^1\backslash\{\pm1\}$ and any complex analytic subvariety $U\subset S$, the restriction $\sigma_{\operatorname{Dol}}|_U$ is holomorphic if and only if $\lambda\cdot\sigma_{\operatorname{Dol}}|_U$ is also an isomonodromic deformation.

The conjecture describes when the $\mathbb C^*$-action preserves isomonodromic deformations. The corresponding assertion is known for $\lambda\in S^1$ by the results mentioned in the source, while the conjectured statements for general real $\lambda$ and the stated equivalence remain open.