The wheel-graph matching-polytope degree conjecture

For an integer $d\geq 4$, let $W_d$ be the wheel graph with rim vertices $1,\ldots,d-1$ forming a cycle and hub vertex $d$ adjacent to every rim vertex. Let $\mathcal{M}_{W_d}$ be its matching polytope, let $I_{\mathcal{M}_{W_d}}$ be the associated toric ideal, and let $\omega(I_{\mathcal{M}_{W_d}})$ denote the maximal degree of a minimal generator.

Wheel-graph degree conjecture. One has

$$\omega(I_{\mathcal{M}_{W_d}})= \begin{cases} 2 & \text{if $d$ is even},\\ 4 & \text{if $d$ is odd}. \end{cases}$$

The source notes that the odd case has lower bound $4$ via odd subdivisions, while computations establish the value $2$ for $W_4$ and $W_6$; the formula for all $d\geq4$ remains open.