The Okrasa–Rzążewski conjecture on indecomposable graph cores

Let $H$ be a connected core on $k\geq 3$ vertices. Write $E_H$ for the edge relation of $H$, let $\langle E_H\rangle$ denote the relational clone generated by $E_H$, and let $\text{NEQ}_k$ be the $k$-ary disequality relation. Okrasa–Rzążewski conjecture. $H$ is indecomposable if and only if

$$\text{NEQ}_k\in\langle E_H\rangle.$$

The paper verifies this conjecture for graphs with at most $7$ vertices. Whether it holds for graphs with more than $7$ vertices remains open.