Whiskered-cycle depth formula conjecture for square-free powers

Let $W(C_m)$ be the graph obtained by attaching a whisker to each vertex of the cycle $C_m$, let $I(W(C_m))$ be its edge ideal in the corresponding polynomial ring $R=\mathbb{K}[x\mid x\in V(W(C_m))]$, and let $1\leq k\leq m$. Whiskered-cycle depth conjecture.

$$\mathrm{depth}(R/I(W(C_m))^{[k]})=\begin{cases}m+k-1 & \text{if } k=1,\ldots,\left\lfloor\frac{m}{2}\right\rfloor,\\ 2k-1 & \text{if } k=\left\lfloor\frac{m}{2}\right\rfloor+1,\ldots,m.\end{cases}$$

The source derives the cases $k=1$ and $k=2$ and bases the formula on computational evidence; the general statement remains open.