= Solution
{c}
For axial compression along $z$, the ordering is reversed: $d_{xy}$ lies below the $d_{xz},d_{yz}$ doublet, and $d_{x^2-y^2}$ lies below $d_{z^2}$ within $e_g$. One electron occupies $d_{xy}$ and the other occupies either $d_{xz}$ or $d_{yz}$, with parallel spins in the weak-distortion regime. The orbital degeneracy is therefore two, with $S=1$.
The retained orbital degeneracy allows further symmetry lowering and explains why this compressed arrangement does not provide the fully quenched equilibrium of the elongated case in the specified simple model.
\Image[/past-exam-of-the-condensed-matter-physics-major-option-course-of-the-university-of-oxford/2019-orbital-levels-and-order.png]
{title=Octahedral d-level splitting and a continuous Jahn-Teller order parameter}
{description=Original small-distortion schematic. The three level panels show the regular, elongated, and compressed one-electron orbital orderings; colored dots mark the high-spin $d^2$ occupation, including one representative choice within a degenerate doublet. Level spacings are illustrative. The last panel plots a normalized square-root mean-field <order parameter>, not measured data or a claim that its exponent is fixed by the source.}
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