Vertex-multiplication conjecture for the matrix T

Let $T$ be the $11\times 11$ matrix displayed in the paper, and let $\tilde{T}$ be any vertex multiplication of $T$. Vertex-multiplication conjecture. One should have

$$\frac{\lambda_{n-1}(\tilde{T})}{|V(\tilde{T})|} \ge -\frac{1}{3}.$$

This conjecture concerns the structured matrices arising from the proposed extremal configurations; the supplied text does not establish it.