Convergence of non-abelianization under Novikov parameter substitution

Determine whether the non-abelianized local system obtained via spectral networks over the Novikov field converges upon the substitution T = e^{-1/ħ} for sufficiently small positive ħ when the base field is ℂ.

Background

The paper constructs a non-abelianization map that transforms rank-1 local systems on a spectral curve L into rank-K local systems on the base C \ D over the Novikov field Λ, using spectral networks and sheaf quantization techniques. Beyond this formal setup, the author raises an analytic convergence question: whether the resulting Λ-local systems are actually convergent after substituting the Novikov parameter by T = e^{-1/ħ} with small ħ.

Establishing such convergence would link the formal Novikov-valued constructions to analytic structures relevant in exact WKB analysis and complex geometry, but it remains conjectural.