Nonvanishing of the cotangent-weight sum at Möbius fixed points

Prove that the quantity \(\ell(z^*)=\sum_k \omega_k\) is nonzero for every simple quadrilateral, where \(z^*\) is a fixed point of the Möbius involution exchanging opposite corners and \(\omega_k\) are the cotangent weights of the fan triangulation at \(z^*\).

Background

The paper identifies the two fixed points $z^*$ of the Möbius involution that exchanges opposite corners of a quadrilateral and proves that each is a critical point of the Dirichlet-energy function when the optimal inserted value $u^*$ is defined there. The optimal value is defined through the positive quantity $\ell(z)=\sum_k\omega_k$, which is guaranteed to be nonzero for insertion points in the kernel $K$.

The authors note that a fixed point $z^*$ can lie outside $K$, including for convex quadrilaterals. They also show that a degenerate configuration involving collinear inverted vertices implies $\ell(z^*)=0$, but state that this does not exhaust the possible mechanisms for vanishing. The unresolved conjecture is that vanishing cannot occur for any simple quadrilateral.