Depth lower bound for powers of the path ideal of a cycle graph

Let $S=K[x_1,\ldots,x_n]$, let $J_{n,m}$ be the path ideal of the cycle graph on $n$ vertices with paths of length $m$, let $d$ be the parameter defined by the paper for $n$ and $m$, and let $t\geq 1$. Depth conjecture.

$$\operatorname{depth}(S/J_{n,m}^t)\geq d-1.$$

The preceding theorem establishes the corresponding upper bound for all $t\geq t_0$, so the conjecture would determine the depth as $d-1$ for those powers. It is proposed on the basis of computer experiments in CoCoA; the general lower bound for all powers remains open.