The fundamental gap conjecture for triangular domains

Let $T\subset\mathbb{R}^2$ be a triangular domain, and let $\xi(T)$ denote the difference between its first two Dirichlet eigenvalues.

Triangular fundamental-gap conjecture.

$$\xi(T)\geq\frac{64\pi^2}{9},$$

where equality holds if and only if $T$ is equilateral.

The equilateral triangle has explicitly computable eigenvalues and eigenfunctions, giving the value $\xi(T)=64\pi^2/9$. This conjecture would follow immediately from the extremum and boundary-monotonicity conjecture, and the source records it as an open gap conjecture for triangular domains.