Isomonodromic deformations of irregular connections and stability of bundles

Let $G$ be a reductive affine algebraic group defined over $\mathbb C$, and let $\nabla_0$ be a meromorphic $G$-connection on a holomorphic $G$-bundle $E_0$, over a smooth complex curve $X_0$, with polar locus $P_0 \subset X_0$. We assume that $\nabla_0$ is irreducible in the sense that it does not factor through some proper parabolic subgroup of $G$. We consider the universal isomonodromic deformation $(E_t\to X_t, \nabla_t, P_t)_{t\in \mathcal{T}}$ of $(E_0\to X_0, \nabla_0, P_0)$, where $\mathcal{T}$ is a certain quotient of a certain framed Teichm\"uller space we describe. We show that if the genus $g$ of $X_0$ satisfies $g\geq 2$, then for a general parameter $t\in \mathcal{T}$, the $G$-bundle $E_t\to X_t$ is stable. For $g\geq 1$, we are able to show that for a general parameter $t\in \mathcal{T}$, the $G$-bundle $E_t\to X_t$ is semistable.