Kao–Lai–Osting conjecture for the second eigenvalue of the torus

Let $T$ be the torus, and let $\Lambda_2(T)$ denote its second topological eigenvalue. Kao–Lai–Osting conjecture.

$$\Lambda_2(T)=\frac{8\pi^2}{\sqrt{3}}+8\pi.$$

This conjecture concerns the sharp value of the second eigenvalue among metrics on the torus and is the target of the paper's improved upper-bound result. Its resolution is not established in the supplied text.