In continuation of the previous question, what is a strict upper bound on the chromatic number of the square of a bipartite graph?

I think the chromatic number number of the square of the bipartite graph with maximum degree $\Delta=2$ and a cycle is at most $4$ and with $\Delta\ge3$ is at most $\Delta+1$. This is because the edge set of a connected bipartite graph consists of the edges of a union of trees and a edge disjoint union of even cycles (with or without chords). Now, the square of a cycle requires at most $4$ colors, and the square of a tree requires at most $\Delta+1$ colors. Thus, the required number of colors is $\Delta+1$ in the latter case. Am I right here? Any counterexamples? Thanks beforehand.