Strong Pólya–Szegő conjecture for finite spectral determination of regular polygons

Let $n\geq 3$, let $\Omega$ be a convex $n$-gon, and let $P$ be a regular $n$-gon. Write $\lambda_i(\Omega)$ for the $i$th eigenvalue of the Dirichlet Laplacian on $\Omega$, and let $N=N(n)$ be the integer from Theorem 3. Strong Pólya–Szegő conjecture. The number $N=N(n)$ in Theorem 3 may be taken equal to $1$; equivalently, if the first eigenvalue of a convex $n$-gon coincides with that of a regular $n$-gon, then the convex $n$-gon is congruent to that regular $n$-gon. The preceding theorem establishes finite spectral determination for some $N(n)$ without an area-normalization assumption. The conjecture strengthens the area-constrained Pólya–Szegő conjecture, and the source gives no resolution.