Trace conjecture for non-nef Hodge bundles

Let $\mathcal{X}/C$ be an abelian scheme and suppose it does not admit an isotrivial abelian subscheme. If

$$\mu_{\min}(f_*\Omega^1_{\mathcal{X}/C})<0,$$

then for some finite étale cover $C'\twoheadrightarrow C$, the $K(C')/k$-trace morphism

$$\tau_{K(C')/k}: (\operatorname{Tr}_{K(C')/k}\mathcal{X}_{K(C')})_{K(C')}\longrightarrow X_{K(C')}$$

is nonzero and cannot be descended to $k$. Trace conjecture for non-nef Hodge bundles. Thus non-nefness of the Hodge bundle should force a nontrivial, non-descendable trace after a finite étale cover. This would make the relationship between Hodge-bundle non-nefness and trace nontriviality precise; the claim is presented as a conjecture and its resolution is not stated in the source.