Vertex-critical graphs far from edge-criticality

Let $r$ be any positive integer. We prove that for every sufficiently large $k$ there exists a $k$-chromatic vertex-critical graph $G$ such that $\chi (G-R)=k$ for every set $R\subseteq E(G)$ with $\mathrm{\mid }R\mathrm{\mid }\le r$. This partially solves a problem posed by Erdős in 1985, who asked whether the above statement holds for $k\ge 4$.