Five-letter complete bifurcate-tree conjecture

A square-free word is bifurcate over an alphabet $\mathcal{A}$ if it has at least one square-free extension at every position. A complete bifurcate tree is an infinite rooted tree of bifurcate words in which every word of length $n$ has $n+1$ descendants, corresponding to extensions at the different positions, and every descendant remains bifurcate.

Five-letter complete bifurcate-tree conjecture. There exists a complete bifurcate tree over an alphabet of size $5$.

The paper proves that a complete bifurcate tree exists over alphabets of size at least $12$ and states that the five-letter bound is tight in the sense that a smaller alphabet cannot suffice. The supplied text does not report a resolution of the five-letter existence claim.