NP-completeness of deciding whether the covering number is at most k

Let $G$ be a graph, and let $\sigma(G)$ denote the minimum number of equivalence relations needed to cover the line graph structure under the paper's definition. For an integer $k\geq 3$, the NP-completeness conjecture. It is NP-complete to decide whether or not

$$\sigma(G)\leq k.$$

The preceding results establish NP-completeness for $k=3$ and $k=4$; the conjecture asserts that this remains true for every larger integer $k$.