The square-free word conjecture for sequences of ternary-or-larger alphabets

An alphabet is a finite set. Let $(\mathcal{A}_i)_{i\ge1}$ be a sequence of alphabets, and say that a word $w=w_1\ldots w_n$ respects this sequence when $w_i\in\mathcal{A}_i$ for every $i$. A word is square-free if it contains no factor of the form $uu$ for a nonempty word $u$.

Square-free word conjecture. Given a sequence of alphabets $(\mathcal{A}_i)_{i\ge1}$ with $|\mathcal{A}_i|\ge3$ for all $i$, there exists an infinite square-free word that respects $(\mathcal{A}_i)_{i\ge1}$.

The source states this as a reformulation of the Thue list-number conjecture for the infinite path. It is presented as an open conjectural formulation, while the paper's main result concerns the analogous existence theorem for binary alphabets and cube-free words.