Ampleness conjecture for trace-zero bundles over Picard-rank-one Fano manifolds

Let $f:X\to Y$ be a finite surjective morphism of degree $d\geq 2$ between projective manifolds $X$ and $Y$ of dimension $n$. Define the trace-zero bundle ${\cal E}$ by

$$f_*({\cal O}_X)={\cal E}^*\oplus {\cal O}_Y.$$

Fano ampleness conjecture. If $Y$ is Fano with $\rho(Y)=1$, then ${\cal E}$ is ample, and is spanned in most cases. This is presented as the guideline for the paper’s section on coverings over Del Pezzo manifolds; the source does not state a resolution.