List steady-word conjecture for four-letter alphabets

A word $W=w_1w_2\cdots w_n$ is steady if it is square-free and remains square-free after deleting any single letter. Let $n$ be a positive integer, and let $\mathcal{A}_1,\mathcal{A}_2,\ldots,\mathcal{A}_n$ be alphabets, each of size $4$.

List steady-word conjecture. There exists a steady word $W=w_1w_2\cdots w_n$ such that $w_i\in\mathcal{A}_i$ for every $i=1,2,\ldots,n$.

The paper proves the corresponding statement for alphabets of size at least $7$. The analogous conjecture for pure square-free words is discussed separately and is not this claim; no resolution of the four-letter steady-word version is supplied here.