K_l-critical graph conjecture

For an integer $\ell\geq 2$, call a graph $G$ $K_{\ell}$-critical if it contains a copy of $K_{\ell}$, is vertex-critical, and removing the vertex set of any copy of $K_{\ell}$ reduces its chromatic number by $\ell$. $K_{\ell}$-critical graph conjecture. If $G$ is $K_{\ell}$-critical for some $\ell\geq 2$, then $G$ is a complete graph. This conjecture generalizes the double-critical graph conjecture and is presented as an open strengthening motivated by counterexamples to the Erdős–Lovász–Tihany conjecture.