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 $>r$?

Dirac's conjecture was proved, for $k=5$: There exists a graph $G$ with chromatic number $5$, such that every vertex is critical, yet every critical set of edges has size $>1$, or in other words: has no critical edge.

[Br92] Brown, Jason I., A vertex critical graph without critical edges. Discrete Math. (1992), 99--101