This is a weakening of [that][1] question.

Let $G=(V,E)$ be a simple, undirected, connected graph, let $n=|V|$ and assume $n\geq 3$. Suppose that whenever you contract $2$ edges, the chromatic number decreases.

Does this imply that $G$ contains a clique of $n-1$ vertices?

  [1]: https://mathoverflow.net/questions/247738/doubly-edge-contraction-critical-graphs