  The mass sensitivity of the vibration-rotation-inversion transitions of
H$_3{}^{16}$O$^+$, H$_3{}^{18}$O$^+$, and D$_3{}^{16}$O$^+$ is investigated
variationally using the nuclear motion program TROVE~\citep{TROVE:2007}. The
calculations utilize new high-level \textit{ab initio} potential energy and
dipole moment surfaces. Along with the mass dependence, frequency data and
Einstein A coefficients are computed for all transitions probed. Particular
attention is paid to the $\Delta|k|=3$ and $\Delta|k-l|=3$ transitions
comprising the accidentally coinciding $|J,K\!=\!0,v_2\!=\!0^+\rangle$ and
$|J,K\!=\!3,v_2\!=\!0^-\rangle$ rotation-inversion energy levels. The newly
computed probes exhibit sensitivities comparable to their ammonia and methanol
counterparts, thus demonstrating their potential for testing the cosmological
stability of the proton-to-electron mass ratio. The theoretical TROVE results
are in close agreement with sensitivities obtained using the nonrigid and rigid
inverter approximate models, confirming that the \textit{ab initio} theory used
in the present study is adequate.
