A spectral bound for the chromatic number involving the number of edges

Let $G$ be a non-empty graph with $m$ edges, chromatic number $\chi$, and least adjacency eigenvalue $\lambda_n$. The spectral edge-count conjecture.

$$\chi(\chi-1) \leq (m+1-\lambda_n^2)+\sqrt{(m+1-\lambda_n^2)^2-4(\lambda_n^2-1)(\lambda_n^2-m)}.$$

This conjecture is motivated by replacing the order parameter in the Fan–Yu–Wang bound with the number of edges, the chromatic-number term by $\chi(\chi-1)$, and the least eigenvalue by $-\lambda_n^2$. The supplied text presents it as a conjecture and gives no resolution.