Bound the proper no-rainbow-P4 parameter for connected cubic graphs

Prove or disprove that every connected cubic graph of order n has a maximum of at most n/2+1 colors in a proper vertex-coloring with no rainbow path on four vertices.

Background

The paper develops lower bounds for colorings of trees without rainbow paths and discusses analogous questions for broader graph families. In particular, it points to connected cubic graphs as a natural family for extending the four-vertex-path theory beyond trees.

The stated conjecture concerns the proper coloring parameter: it asserts an upper bound of n/2+1 on the number of colors in a proper coloring that avoids a rainbow P4. The paper does not resolve this conjecture.