Precoloring extension for sufficiently generic near-Eulerian triangulations

Let $\ell$ be a positive even integer. A planar near-Eulerian-triangulation is a planar graph with triangular bounded faces and even vertex degrees. Let $G$ have outer face bounded by a cycle $C$ of length $\ell$, and let $U$ be an independent set disjoint from $C$ such that $G-U$ is bipartite. Let $S$ be a set of $d$ faces of $G-U$, each of length at least six. A 4-coloring of $C$ is viable if it satisfies the viability condition defined in the source.

Generic precoloring-extension conjecture. For every positive even integer $\ell$, there exists an integer $d$ such that, if no 4-cycle in $G-U$ separates a face of $S$ from the outer face, and for distinct faces $s_1,s_2\in S$ the distance between $s_1$ and $s_2$ in $G-U$ is at least $d$ and no closed walk of length less than $\ell$ in $G-U$ separates both $s_1$ and $s_2$ from the outer face, then any viable 4-coloring of $C$ extends to a 4-coloring of $G$.

This conjecture proposes a sufficient condition for extending viable boundary colorings in sufficiently generic instances, and is presented as a step toward the bounded-boundary-length precoloring-extension conjecture. The definition of viability is deferred to the source, and no resolution is given.