The corner-factor conjecture for Neumann jump determinants

Let $P$ be a compact simply connected surface of constant curvature with piecewise geodesic boundary, whose vertices have interior angles

$$\pi\alpha_1,\ldots,\pi\alpha_N,\qquad 0<\alpha_j<2.$$

Let $\widehat P=P\cup_{\partial P}P$ be the mirror double, and use the Friedrichs Laplacian on $\widehat P$ and the Friedrichs Dirichlet Laplacian on $P$. Define the BFK quotient by

$$\mathcal J_{\rm BFK}(P):= \frac{\operatorname{det}'\Delta_{\widehat P}/\operatorname{Area}(\widehat P)} {\bigl(\operatorname{det}\Delta_{P,D}\bigr)^2}.$$

Corner-factor conjecture. There is a canonical determinant $\operatorname{det}_{\angle}'\mathcal N_{\partial P}$ of the Neumann jump operator on a piecewise real-analytic curve with finitely many corners for which the BFK formula remains valid. In the mirror-double setting,

$$\frac{\operatorname{det}_{\angle}'\mathcal N_{\partial P}}{\operatorname{length}(\partial P)}=\mathcal J_{\rm BFK}(P)=\frac12\prod_{j=1}^{N}\alpha_j^{-1/2}.$$

This proposes a canonical extension of the Neumann jump determinant and the BFK formula to cornered cuts, where the jump operator is not a classical pseudodifferential operator and ordinary zeta regularization may require modification. The BFK quotient itself is well defined when the Laplacian determinants exist; the existence and normalization of the corner determinant remain open.