Kouider–Lonc's balanced path-decomposition conjecture

Let $\ell$ be a positive integer, and let $P_\ell$ be the path with $\ell$ edges. A balanced $P_\ell$-decomposition of a graph $G$ is a path decomposition in which every vertex is the end-vertex of exactly two paths.

Kouider–Lonc's conjecture. If $G$ is $2\ell$-regular, then $G$ admits a balanced $P_\ell$-decomposition.

This strengthens the tree-decomposition conjecture for paths. Kouider and Lonc proved the statement when the girth satisfies $g\geq(\ell+3)/2$, while the paper verifies it for paths of length $4$; the general assertion is not resolved in the supplied text.