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Global generation for unitary local systems on curves

Prove that for g≥3 and a unitary representation \rho: \pi_1(\Sigma_g)\to U(r), for a generic complex structure X on \Sigma_g, the underlying vector bundle \mathscr{E} of the corresponding flat bundle (\mathscr{E},\nabla) satisfies that \mathscr{E}\otimes \omega_X is generated by global sections.

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Background

This strengthening of the ggg conjecture predicts full global generation (not just generic) for twists of unitary bundles on generic curves, a property that underpins proofs of maximal monodromy statements in the paper.

References

Conjecture Let $g\geq 3$, and let $$\rho: \pi_1(\Sigma_{g})\to U(r)$$ be a unitary representation. Fix a generic complex structure $X$ on $\Sigma_{g}$, and let $(\mathscr{E}, \nabla)$ be the corresponding flat bundle. Then $\mathscr{E}\otimes \omega_{X}$ is generated by global sections.

— Motives, mapping class groups, and monodromy (2409.02234 - Litt, 3 Sep 2024) in Conjecture, Section 6.3.2