Polynomial-time precoloring extension for bounded outer faces

A planar near-Eulerian-triangulation is a planar graph in which every bounded face is a triangle and every vertex has even degree; a precoloring assigns colors from a set of four colors to the vertices incident with the outer face. For a positive integer $\ell$, consider instances whose outer face has length at most $\ell$.

Polynomial-time precoloring-extension conjecture. For every positive integer $\ell$, there is a polynomial-time algorithm that, given a planar near-Eulerian-triangulation $G$ with outer face of length at most $\ell$ and a precoloring $\varphi$ of the vertices incident with the outer face, decides whether $\varphi$ extends to a 4-coloring of $G$.

This conjecture extends the proved linear-time result for outer faces of length at most five and would resolve the disk case for every fixed boundary length. The source gives no resolution beyond these bounded-length results.