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Stability under isomonodromy for unitary flat bundles

Establish that for a marked curve (X,D) of genus g≥2 and an irreducible unitary flat bundle (\mathscr{E},\nabla) with regular singularities along D, the underlying holomorphic bundle \mathscr{E} becomes (semi)stable after a general isomonodromic deformation.

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Background

This conjecture is a Riemann–Hilbert type expectation for behavior of flat bundles under isomonodromic deformation: unitary monodromy should force the underlying vector bundle to acquire stability generically after small deformation of complex structure.

References

Conjecture Let $(X,D)$ be a marked curve of genus $g$ at least $2$, and $(\mathscr{E},\nabla)$ a flat bundle on $X$ with regular singularities along $D$, and irreducible unitary monodromy. Then after a general isomonodromic deformation, $\mathscr{E}$ is (semi)stable.

— Motives, mapping class groups, and monodromy (2409.02234 - Litt, 3 Sep 2024) in Conjecture, Section 6.4