Hamiltonian paths in the permutation digraphs $P(n,n-2)$

For $1\le k<n$, let $P(n,k)$ be the directed overlap graph whose vertices are the $k$-permutations of $[n]$ and whose arcs are the $(k+1)$-permutations. Isaak proved that $P(n,n-2)$ has no directed Hamiltonian cycle for $n\ge 4$ and asked whether it nevertheless has a directed Hamiltonian path. We answer this question affirmatively by showing that $P(n,n-2)$ has a Hamiltonian path.