Cohen–Macaulay linear-resolution conjecture for complexes from squared paths

Let $L_n^2$ be the graph on $[1,n]$ with edges $\{i,i+1\}$ for $i=1,\ldots,n-1$ and $\{i,i+2\}$ for $i=1,\ldots,n-2$. Let $\Delta_k^t(L_n^2)$ be the associated complex, and let its dual be the Alexander dual complex. Cohen–Macaulay linear-resolution conjecture for $\Delta_k^t(L_n^2)$. Both $\Delta_k^t(L_n^2)$ and its dual have Cohen–Macaulay Stanley–Reisner rings with linear resolutions. The surrounding theorem and proof establish the corresponding assertion for $\Delta_2^t(L_n^2)$, but the supplied statement is written with general $k$ and the text does not resolve whether that generality is intended or prove it.