The triangle-count conjecture under a spectral-radius condition

Let $G$ be a graph with $m$ edges, and let $(G)$ denote its spectral radius. A triangle is a copy of $K_3$, and suppose that no vertex of $G$ is contained in all triangles of $G$.

Triangle-count conjecture. If

$$(G)>\sqrt{m},$$

then $G$ contains at least

$$\sqrt{m}-O(1)$$

triangles, where $O(1)$ is bounded independently of $m$.

This conjecture aims to double, asymptotically, the triangle lower bound of Ning and Zhai under the same spectral constraint, after excluding the case in which one vertex lies in every triangle. The accompanying construction shows that the order of the proposed bound is asymptotically tight.