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 $\dim X=n\ge 3$. 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.