Tevelev degrees in Gromov-Witten theory are defined whenever there are virtually a finite number of genus g maps of fixed complex structure in a given curve class beta through n general points of a target variety X. These virtual Tevelev degrees often have much simpler structure than general Gromov-Witten invariants. We explore here the question of the enumerativity of such counts in the asymptotic range for large curve class beta. A simple speculation is that for all Fano X, the virtual Tevelev degrees are enumerative for sufficiently large beta. We prove the claim for all homogeneous varieties and all hypersurfaces of sufficiently low degree (compared to dimension).
