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The mass sensitivity of the vibration-rotation-inversion transitions ofH$_3{}^{16}$O$^+$, H$_3{}^{18}$O$^+$, and D$_3{}^{16}$O$^+$ is investigatedvariationally using the nuclear motion program TROVE~\citep{TROVE:2007}. Thecalculations utilize new high-level \textit{ab initio} potential energy anddipole moment surfaces. Along with the mass dependence, frequency data andEinstein A coefficients are computed for all transitions probed. Particularattention is paid to the $\Delta|k|=3$ and $\Delta|k-l|=3$ transitionscomprising the accidentally coinciding $|J,K\!=\!0,v_2\!=\!0^+\rangle$ and$|J,K\!=\!3,v_2\!=\!0^-\rangle$ rotation-inversion energy levels. The newlycomputed probes exhibit sensitivities comparable to their ammonia and methanolcounterparts, thus demonstrating their potential for testing the cosmologicalstability of the proton-to-electron mass ratio. The theoretical TROVE resultsare in close agreement with sensitivities obtained using the nonrigid and rigidinverter approximate models, confirming that the \textit{ab initio} theory usedin the present study is adequate.

Accurate prediction of H$_3$O$^+$ and D$_3$O$^+$ sensitivity coefficients to probe a variable proton-to-electron mass ratio