Existence of arbitrarily long extremal square-free permutations

An extremal square-free permutation is a square-free permutation for which inserting any letter in any position introduces a square. The extremal-permutation conjecture. There exist arbitrarily long extremal square-free permutations. Extremal permutations are known to exist at lengths $17$ and $21$, but not at any other length $n\leq 22$; the assertion remains open.