Cycle depth formula conjecture for square-free powers

Let $C_n$ be a cycle of length $n$, let $I(C_n)$ be its edge ideal in the corresponding polynomial ring $R=\mathbb{K}[x\mid x\in V(C_n)]$, and let $\nu(C_n)$ denote its matching number. Cycle depth conjecture. For the indicated values of $k$,

$$\mathrm{depth}(R/I(C_n)^{[k]})=\begin{cases}\left\lceil\frac{n}{3}\right\rceil+k-1 & \text{if } k=2,\ldots,\left\lceil\frac{n}{3}\right\rceil,\\ 2k-1 & \text{if } k=\left\lceil\frac{n}{3}\right\rceil+1,\ldots,\nu(C_n).\end{cases}$$

The cases $k=1$, $k=2$, and $k=\nu(C_n)$ are known in the source, and the case $k=\nu(C_n)-1$ is established for cycles of even length; the conjecture was verified computationally up to $n=15$, while the remaining cases are open.