Gray-code canonical coloring conjecture for powers of paths

Let $P_n^\ell$ be the $\ell$th distance-power of the path on vertices $v_0,\ldots,v_{n-1}$, so that $v_i$ and $v_j$ are adjacent exactly when $|i-j|\le \ell$. Let $\phi$ be the canonical edge-coloring defined using the binary reflected Gray code in Construction~, and let an optimal strong parity edge-coloring mean one using exactly $\hat{p}(P_n^\ell)$ colors.

Gray-code canonical coloring conjecture. The canonical edge-coloring $\phi$ is an optimal strong parity edge-coloring of $P_n^\ell$.

The conjecture asserts that the upper bound obtained from this Gray-code construction is sharp. The paper proves matching upper and lower bounds up to a quantity quadratic in $\ell$ and states that the upper bound is expected to be the truth; no resolution is supplied in the given text.