A note on the proper conflict-free connection number of connected graphs

Let G be an edge-colored graph. A path in G is a conflict-free path if it contains a color used on exactly one of its edges. The graph G is called conflict-free connected if every two distinct vertices is connected by at least one conflict-free path. The graph G is said to be properly edge colored (properly colored for simplifying), meaning an assignment of colors to edges so that no vertex is incident to two edges of the same color.