We determine the full 1/sqrt(lambda) correction to the flat-space Wilson coefficients which enter the AdS Virasoro-Shapiro amplitude in N=4 SYM theory at strong coupling. The assumption that the Wilson coefficients are in the ring of single-valued multiple zeta values, as expected for closed string amplitudes, is surprisingly powerful and leads to a unique solution to the dispersive sum rules relating Wilson coefficients and OPE data. The corresponding OPE data fully agrees with and extends the results from integrability.