point group symmetry at the V sites from cubic to tetragonal. The compression along a single axis corresponds to the $C_{11}-C_{12}$ stability condition of the elastic constants $C_{ij}$ for cubic crystals [@Cowley76]. Consequently, the triplet degeneracy of the $t_{2g}$ orbital is partially lifted. One of the vanadium electrons occupies the energetically lower-lying $d_{xy}$ orbital, while the other electron is shared among the degenerate $d_{xz}$ and $d_{yz}$ orbitals. The $d_{xy}$ orbitals on the pyrochlore lattice order in a way that their lobes point to neighbouring vanadium ions in the $ab$ plane, along $\left[110\right]$ and $\left[1\overline{1}0\right]$ directions [@Tcherny04]. The occupation of the $d_{xz}$ and $d_{yz}$ orbitals is an unresolved problem arising from the competition of three different interaction mechanisms [@Khomskii2014]. Namely the coupling of the orbital to the elastic strain via the cooperative Jahn-Teller effect, the Kugel-Khomskii interaction between neighbouring vanadium ions, and the spin-orbit coupling. Depending on which interaction is assumed to dominate, different orbital ordering patterns are obtained.

A prominent theoretical model, given by Tsunetsugu and Motome [@Tsunetsugu03], is based on the Kugel-Khomskii interaction between the vanadium ions, and results in a ferro-orbital (antiferro-orbital) ordering of the $d_{xy}$ ($d_{xz}$, $d_{yz}$) orbitals along the \[110\] (\[101\], \[011\]) directions in the pyrochlore 