Generic vanishing on homogeneous spaces in arbitrary characteristic

Let $X$ be a proper homogeneous space for a connected algebraic group $G$ over an algebraically closed field. For locally closed smooth affine subvarieties $W,Z\subset X$, we show that $$(-1)^{\dim X-\dim W+\dim Z}\chi(gW\cap Z)\geq 0$$ for generic $g\in G$. This extends the characteristic-zero theorem of Sch\"urmann--Simpson--Wang. Over finite fields, our methods give a trace-function identity on a dense open subset of $G$ and a Lang--Weil estimate for the non-generic locus.