On $P$-crucial square-free permutations

A permutation is square-free if it does not contain two consecutive factors of length two or more that are order-isomorphic. A square-free permutation of length $n$ is $P$-crucial, where $P$ is a subset of $\{0,1,\ldots,n\}$, if any of its extensions in any position from the set $P$ contains a square. In 2015, Gent, Kitaev, Konovalov, Linton and Nightingale initiated the study of $P$-crucial square-free permutations. In particular, they showed that $\{0,1,n-1,n\}$-crucial square-free permutations of length $n$, where $n\leq 22$, exist if and only if $n=17$ or $n=21$. In this work, we prove that for any $m\geq 2$ there exists a $\{0,1,8m+4,8m+5\}$-crucial square-free permutation of length $8m+5$.