Polynomial-time k-colouring conjecture for graphs with maximal local edge-connectivity k

Let $k\geq 4$ be fixed, and let $G$ be a graph with maximal local edge-connectivity $k$, meaning that the maximum local edge-connectivity over all pairs of vertices of $G$ is $k$. A $k$-colouring of $G$ is a proper vertex colouring using at most $k$ colours.

Polynomial-time k-colouring conjecture. There is a polynomial-time algorithm that, given $G$, finds a $k$-colouring of $G$, or determines that none exists.

The conjecture concerns the unresolved complexity of $k$-colouring for graphs with maximal local edge-connectivity $k$ when $k\geq 4$, following polynomial-time results for narrower connectivity classes and NP-completeness results for related colouring problems.