Depth formula conjecture for squarefree powers of wheel edge ideals

Let $\mathcal W_n$ be the wheel graph on $n+1$ vertices, let $I(\mathcal W_n)$ be its edge ideal in $R=\mathbbk[x_1,\ldots,x_n,v]$, and let $\nu(\mathcal W_n)$ denote its matching number. For integers $n\ge3$ and $1\le q\le\nu(\mathcal W_n)$, consider the $q$-th squarefree power $I(\mathcal W_n)^{[q]}$. Depth conjecture for wheel squarefree powers. For all such $n$ and $q$,

$$\operatorname{depth}(R/I(\mathcal W_n)^{[q]})=2q-1,$$

equivalently,

$$\operatorname{pd}(R/I(\mathcal W_n)^{[q]})=n-2q+2.$$

The conjecture is suggested by the complete computations reported for wheels with $5\le n\le9$ in the relevant low-power cases, together with the top-power cases for $n=5,\ldots,8$. It would give a uniform depth and projective-dimension formula throughout the full range of squarefree powers, beyond the computational evidence currently recorded.