Forward-or-backward certifying-path conjecture

Let $G$ be a connected $k$-chromatic graph. In a $k$-coloring, a certifying path is a path whose vertices meet the required distinct color classes, and call such a path forward or backward when consecutive colors increase or decrease cyclically, respectively. Forward-or-backward certifying-path conjecture. Every $k$-chromatic connected graph different from $C_7$ admits a $k$-coloring such that every vertex is the end of a forward or backward certifying path. This is a strengthening of the colorful-path conjecture: the source notes that the assertion is known in certain cases, but leaves the strengthened statement open in general.