The Hitchin morphism for certain surfaces fibered over a curve

The Chen-Ng\^o Conjecture predicts that the Hitchin morphism from the moduli stack of $G$-Higgs bundles on a smooth projective variety surjects onto the space of spectral data. The conjecture is known to hold for the group $GL_n$ and any surface, and for the group $GL_2$ and any smooth projective variety. We prove the Chen-Ng\^o Conjecture for any reductive group when the variety is a ruled surface or (a blowup of) a nonisotrivial elliptic fibration with reduced fibers. Furthermore, if the group is a classical group, i.e. $G \in \{SL_n,SO_n,Sp_{2n}\}$, then we prove the Hitchin morphism restricted to the Dolbeault moduli space of semiharmonic $G$-Higgs bundles surjects onto the space of spectral data.