Quaternary bifurcate-chain 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 single-letter extension is obtained by inserting one letter at one position. A quaternary bifurcate word is a bifurcate word over an alphabet of size $4$.

Quaternary bifurcate-chain conjecture. There exists an infinite sequence of quaternary bifurcate words $W_1,W_2,\ldots$ such that $W_{i+1}$ is a single-letter extension of $W_i$ for every $i=1,2,\ldots$.

The paper proves a much stronger complete bifurcate-tree property over alphabets of size at least $12$, while the quaternary chain assertion remains open in the supplied text.