A Higher dimensional log Riemann--Hurwitz inequality and rigidity of covers

Let $\overline{Z}$ be a smooth projective variety with an simple normal crossing divisor $D$ such that $Ω_{\overline{Z}}^1(\log D)$ is nef. We prove that if $Y\subset Z:=\overline Z\setminus D$ is a smooth closed subvariety with nonzero Euler characteristic, and $P$ is a perverse sheaf on $Y$ with f