Finiteness conjecture for eigenvalues of the Neumann–Poincaré operator on planar corners

Let $\Gamma$ be a piecewise $C^3$ Jordan curve with finitely many corners, whose angles lie strictly between $0$ and $\pi$. Let $K \colon H^{1/2}(\Gamma) \to H^{1/2}(\Gamma)$ be the Neumann–Poincaré operator. Finiteness conjecture. The operator $K$ has finitely many eigenvalues. This is an outstanding question for planar domains with corners; existing constructions exhibit eigenvalues embedded in the absolutely continuous essential spectrum, but do not suggest the existence of infinitely many eigenvalues.