Clique-number refinement of the positive square-energy conjecture

Let $G$ be a connected graph of order $n$, and let $\omega(G)$ denote its clique number. Clique-number refinement. If

$$\omega(G)\ge3,$$

then

$$s^+(G)\ge n.$$

This conjecture is motivated by the expectation that a large clique number forces large positive square energy. The source notes that it would follow from the strong and unicyclic conjectures if those hold, but the triangle-containing unicyclic case appears to be a principal obstacle.