Spectral triangular-edge counting conjecture

Let $G$ be a graph with $m$ edges, let $\lambda(G)$ denote its spectral radius, and let $t(G)$ denote the number of triangular edges. Spectral triangular-edge counting conjecture. If

$$\lambda(G)\ge\sqrt m,$$

then $G$ has at least $\sqrt m$ triangular edges, unless $G$ is a complete bipartite graph. This strengthens the cited result giving only $\lfloor(\sqrt m-1)/2\rfloor$ triangles under the same spectral condition; the source gives no resolution status.