The upper bound on the strong periodicity of connected graphs under the 2-distance operator

Let $G=(V,E)$ be any connected graph, and let $\rho^2(G)$ denote its strong periodicity under the 2-distance operator.

Upper-bound conjecture. Any connected graph $G=(V,E)$ has

$$\rho^2(G)\leq |V|-3.$$

This conjecture gives a universal linear upper bound on the strong periodicity of connected graphs under the 2-distance operator. The supplied text does not state whether the bound has been proved or refuted.