Maximal multipoint Seshadri constants in positive characteristic

Let $L$ be an ample line bundle on a smooth integral projective variety $X$ over an algebraically closed field of positive characteristic, with $\dim X=n\ge3$. We prove that there is a threshold $r_0=r_0(X,L)$ such that for every integer $r\ge r_0$, the ordinary multipoint Seshadri constant at the geometric generic tuple of $r$ points equals the volume bound $(L^n/r)^{1/n}$. The same conclusion holds in dimension two. This establishes the positive-characteristic form of the qualitative Nagata–Biran–Szemberg assertion at geometric generic tuples.