Conjecture on non-existence of holomorphic isomonodromic deformations of non-nilpotent Higgs bundles

Let $\mathcal T_g$ denote the Teichmüller space of compact Riemann surfaces of genus $g\geq2$, let $\mathcal X/\mathcal T_g$ be its universal family, and let $\mathcal M_{\text{Dol}}(\mathcal X/\mathcal T_g)$ be the relative Dolbeault moduli space parameterizing polystable $\mathrm{SL}(n,\mathbb C)$-Higgs bundles. A Higgs bundle $(E,\bar\partial,\theta)$ in this moduli space is non-nilpotent when its Higgs field is not nilpotent. Non-existence conjecture. For a non-nilpotent Higgs bundle $(E,\bar\partial,\theta)$ in $\mathcal M_{\text{Dol}}(\mathcal X/\mathcal T_g)$, its isomonodromic deformation fails to be holomorphic over $\mathcal T_g$. This conjecture is motivated by the preceding non-existence theorems for generic Higgs bundles and for low-rank non-unitary Higgs bundles; the general non-nilpotent case remains open.