Let $k \ge 4$ and $r\ge 1$. Must there exist a graph $G$ with chromatic number $k$ such that every vertex is critical, yet every critical set of edges has size $>r$?

Lattanzio [La02] proved there exist $k$-critical graphs without critical edges for all $k$ such that $k - 1$ is not prime.

[La02] Lattanzio, John J., A note on a conjecture of {D}irac. Discrete Math. (2002), 323--330