Zaare-Nahandi's semi-perfectness conjecture for multipartite well-covered graphs

Let $G$ be an $s$-partite well-covered graph, meaning that all maximal independent sets have the same size, and suppose that every maximal clique has size $s$. A graph is semi-perfect when its clique covering number $\theta(G)$ equals its independence number $\alpha(G)$.

Zaare-Nahandi's conjecture. $G$ is semi-perfect.

Zaare-Nahandi proposed this conjecture as a characterization related to localizable graphs. It is refuted: the paper constructs counterexamples, using a reduction from $3$-colorability in triangle-free graphs.