Laugesen–Siudeja's simplicity conjecture for the second Dirichlet eigenvalue

Let $T$ be a bounded triangle in the Euclidean plane, and let the Dirichlet eigenvalues of its Laplacian, listed with multiplicity, be

$$0<\lambda_1(T)\leq\lambda_2(T)\leq\cdots.$$

An equilateral triangle is a triangle whose three sides have equal length. Laugesen–Siudeja's conjecture. The second Dirichlet eigenvalue is simple on every non-equilateral triangle; equivalently, if $T$ is not equilateral, then $\lambda_2(T)<\lambda_3(T)$. This conjecture is solved by the paper, which gives a computer-assisted proof and in particular resolves the previously open case of nearly degenerate triangles.