Betti numbers of powers of path ideals of cycles

Let $J_{n,m}=(x_1\cdots x_m,x_2\cdots x_{m+1},\dots ,x_n{x}_1\cdots x_{m-1})$ be the $m$-path ideal of a cycle of length $n\ge 5$ over a polynomial ring $S=k[x_1,\dots ,x_n]$. Let $t\ge 1$ be an integer. We show that $J_{n,m}^t$ has a linear free resolution and give a precise formula for all of its Betti numbers when $m=n-1,n-2$.