Irreducibility from a finite-union description of the spectral base

Ascertain whether, under the hypotheses leading to Proposition 4.19 (existence of finitely many étale-trivializable rank r vector bundles V_i such that B_X^r = ⋃_i h_X^r(V_i)), this finite-union description implies that the spectral base B_X^r is irreducible; alternatively, determine whether this setting can produce examples where B_X^r is not irreducible.

Background

Under suitable vanishing assumptions on symmetric differentials, the authors show every point of B_Xr arises as the Hitchin image of a Higgs field on one of finitely many fixed étale-trivializable bundles, yielding a finite-union covering of B_Xr by such images.

Whether this covering forces irreducibility is not clear; resolving this would clarify the global structure of B_Xr and may yield counterexamples to irreducibility in the K-trivial setting.

References

However, it is unclear whether this description would imply irreducibility of the spectral base in this case. Rather, it may provide us with an example of the spectral base of a $K$-trivial variety where the spectral base is not irreducible.

— The Hitchin morphism for K-trivial varieties  (2604.03217 - Patel et al., 3 Apr 2026) in Remark following Proposition \ref{irred}, Section 4 (A stronger version of the conjecture)