Higher-order corrections to the spin-orbit and spin-spin tensor interactions in HD$^+$

Improved values of hyperfine coefficients related to the electronic spin-orbit and electron-nucleus spin-spin tensor interactions in the HD$^+$ molecular ion are obtained through numerical calculation of relativistic corrections at the $m\alpha^6$ order and radiative corrections at the $m\alpha^7\ln(\alpha)$ order. The theoretical accuracy is improved by more than one order of magnitude. Some deviations with recent high-precision ro-vibrational spectroscopy experiments are observed, in contrast with the good agreement obtained in H$_2^+$.