Existence of arbitrarily long bicrucial permutations of even length

A bicrucial permutation is a square-free permutation such that prepending or appending any single element creates a square. The existence conjecture. There exist arbitrarily long bicrucial permutations of even length. The paper gives constructions in several odd congruence classes and an even example of length $32$, while arbitrarily long even lengths remain to be established.