We revisit the mα6(m/M) order corrections to the hyperfine splitting in the H2+ ion and find a hitherto unrecognized second-order relativistic contribution associated with the vibrational motion of the nuclei. Inclusion of this correction term produces theoretical predictions which are in excellent agreement with experimental data [K. B. Jefferts, Phys. Rev. Lett. 23, 1476 (1969)], thereby concluding a nearly 50-year-long theoretical quest to explain the experimental results within their 1-ppm error. The agreement between the theory and experiment corroborates the proton structural properties as derived from the hyperfine structure of atomic hydrogen. Our work furthermore indicates that, for future improvements, a full three-body evaluation of the mα6(m/M) correction term will be mandatory.