Weighted-graph third-eigenvalue conjecture

For $n\geq 2$, let $\mathcal{S}_n$ be the set of $n\times n$ real symmetric matrices with entries in $[0,1]$, and let $\lambda_{n-1}(M)$ denote the penultimate eigenvalue of $M$. Weighted-graph conjecture. The function $\lambda_{n-1}:\mathcal{S}_n\to\mathbb{R}$ satisfies

$$-\frac{n}{3} \le \inf_{M\in\mathcal{S}_n}\lambda_{n-1}(M).$$

This is the weighted-matrix extension of the extremal third-eigenvalue problem; the supplied text presents it as a conjecture and gives no resolution.