Antunes–Freitas conjecture on three eigenvalues determining a triangle

Let $T_1$ and $T_2$ be triangles, and write $\lambda_i(T_j)$ for their $i$th eigenvalues of the Dirichlet Laplacian. Antunes–Freitas conjecture. If

$$\lambda_i(T_1)=\lambda_i(T_2),\qquad i=1,2,3,$$

then $T_1$ and $T_2$ are identical up to a rigid motion. Antunes and Freitas supplied extensive numerical evidence for this claim, but the source reports that the first, second, and fourth eigenvalues do not uniquely determine a triangle. The conjecture itself remains open.