The ultimate periodicity conjecture for super-monochromatic factorisations

Let $x$ be an infinite word over an alphabet $\mathcal{A}$. A super-monochromatic factorisation of a suffix of $x$ is a factorisation into finite words such that all factors and every ordered concatenation of factors have the same color. Ultimate periodicity conjecture. The word $x$ is ultimately periodic if and only if, for every coloring of the set of finite words over $\mathcal{A}$, $x$ admits a suffix having a super-monochromatic factorisation.

The conjecture links ultimate periodicity of infinite words with colorings that avoid monochromatic factorisations of suffixes, subject to the additional requirement that ordered concatenations remain monochromatic. The paper establishes reductions and verifies the conjecture for squarefree words using the notion of consecutive length; the general statement remains open.