The partition conjecture for k-clique isolating sets

Let $k \geq 3$ be an integer. A $k$-clique isolating set of a graph is a vertex set whose closed neighborhood leaves no copy of $K_k$. The partition conjecture. Every connected graph, except $K_k$, can be partitioned into $k+1$ disjoint $k$-clique isolating sets. This would imply the known upper bound $\iota(G,k) \leq n/(k+1)$ for connected graphs $G \ncong K_k$, and the paper studies this conjecture without resolving it.