Chaigneau's asymptotic conjecture for curvilinear polygons

Let a curvilinear polygon have $n$ vertices with interior angles $\alpha_1,\dots,\alpha_n$. For $\alpha\in(0,2\pi)$, define

$$\overline{k}(\alpha):=\max\left\{k\in\{0\}\cup\mathbb{N}:2k-1<\frac{\pi}{\alpha}\right\}$$

and

$$\mathcal{C}(\alpha):=\begin{cases} \varnothing&\text{if }\alpha\ge\pi,\\ \left\{\sin\frac{\alpha}{2},\sin\frac{3\alpha}{2},\dots,\sin\frac{(2\overline{k}(\alpha)-1)\alpha}{2}\right\}&\text{if }\overline{k}(\alpha)>0. \end{cases}$$

Form the multiset $\widetilde{\mathcal{C}}=\bigcup_{i=1}^n\mathcal{C}(\alpha_i)$, let $K:=\sum_{i=1}^n\overline{k}(\alpha_i)$, arrange its positive elements as $\widetilde{c}_1\le\dots\le\widetilde{c}_K$, and set $\widetilde{c}_k:=1$ for $k>K$. Chaigneau's asymptotic conjecture. For any curvilinear polygon, the asymptotics of the Dirichlet-to-Neumann eigenvalues hold with

$$c_k=\widetilde{c}_k,\qquad k\in\mathbb{N}.$$

This conjecture extends the known asymptotic description to all eigenvalues of an arbitrary curvilinear polygon; the source identifies it as an open question.