Li--Feng--Peng's conjecture on triangular edges in spectral graphs

Let $G$ be a graph with $m$ edges, and let $\lambda(G)$ denote its spectral radius. An edge is triangular if it lies in a triangle. Li--Feng--Peng's conjecture. If

$$\lambda(G)\geqslant\sqrt{m},$$

then $G$ contains at least $\sqrt{m}$ triangular edges, unless $G$ is a complete bipartite graph. This conjecture concerns a spectral extension of the Erdős--Faudree--Rousseau theorem on triangular edges; the source does not provide evidence that it has been resolved.