Conjecture on disconnected decomposition graphs and the Ritt ideal

Let $B$ be a rational function that is not special, meaning that it is neither a Lattès map nor conjugate to $z^{7}$ or to a Chebyshev polynomial $T_n$. Let $\Gamma_B$ be the graph associated with the decompositions of $B$, and let $\Gamma_B\backslash\{B\}$ denote the graph with the vertex corresponding to $B$ removed. The Ritt ideal is the obstruction defined in the paper to the extension of the non-elementary length character from polynomials to rational functions.

Disconnected-graph conjecture. If $\Gamma_B\backslash\{B\}$ is disconnected, then $B$ belongs to the Ritt ideal.

This conjecture proposes a graph-theoretic criterion for detecting rational functions in the Ritt ideal. The paper presents it as an open direction motivated by the obstruction that the Ritt ideal provides to extending the length character.