Caro–Petruševski–Škrekovski conjecture on proper conflict-free chromatic number

Let $G$ be a connected simple finite graph with maximum degree $e=3$. A proper conflict-free coloring is a proper vertex coloring in which every non-isolated vertex has a color appearing exactly once in its open neighborhood; $c_{\rm PCF}(G)$ denotes the minimum number of colors in such a coloring.

Caro–Petruševski–Škrekovski conjecture. The proper conflict-free chromatic number satisfies

$$\chi_{\rm PCF}(G) \leq \Delta+1.$$

Caro, Petruševski, and Škrekovski proved the general upper bound $\chi_{\rm PCF}(G)\leq\lfloor 5\Delta(G)/2\rfloor$; the conjecture would improve this bound for connected graphs of maximum degree at least $3$.