If $G=(V,E)$ is a finite, simple, undirected graph, the Hadwiger number $\eta(G)$ is defined to be the size of the largest complete minor of $G$.

Is there a finite graph $G=(V,E)$ with the following property?

Whenever $v,w\in V$ have distance $2$, the Hadwiger number of the graph obtained by collapsing $v,w$ is strictly larger than $\eta(G)$.