Depth and Stanley depth of powers of the path ideal of a cycle graph. II

Let $J_{n,m}:=(x_1{x}_2\cdots x_m,x_2{x}_3\cdots x_{m+1},\dots ,x_{n-m+1}\cdots x_n,x_{n-m+2}\cdots x_n{x}_1,\dots ,x_n{x}_1\cdots x_{m-1})$ be the $m$-path ideal of the cycle graph of length $n$, in the ring of polynomials $S=K[x_1,\dots ,x_n]$. As a continuation of arxiv:2303.15032v2, we prove several new results regarding $\mathrm{depth}(S\mathrm{/}{J}_{n,m}^t)$ and $\mathrm{sdepth}(S\mathrm{/}{J}_{n,m}^t)$, where $t\ge 1$.