Order \b14 conjecture for boundary operators on surfaces

Let $\Sigma$ be a smooth surface with smooth boundary, endowed with a Riemannian metric. Let $\varkappa$ denote the boundary operator and $D_\Sigma$ the Dirichlet-to-Neumann operator, and set

$$\delta=\varkappa^{-1}D_\Sigma-1.$$

If the boundary of $\Sigma$ is totally geodesic, then $\delta$ is a pseudodifferential operator of order $-4$. If a surface $\Sigma$ is cut by a $1$-manifold $Y$ into two surfaces $\Sigma_L$ and $\Sigma_R$, let $D_{\Sigma_L,\Sigma_R}$ denote the corresponding operator and let $\delta_L$ and $\delta_R$ be the associated operators. Order $-4$ conjecture. One has

$$\frac12(\delta_L+\delta_R)=\frac{1}{2\varkappa_Y}D_{\Sigma_L,\Sigma_R}-1,$$

and this operator is pseudodifferential of order $-4$, without assuming that $Y$ is geodesic.

The claim predicts improved pseudodifferential decay beyond the previously established general order bounds, and reflects the expectation that the singular part of the kernel of $\delta$ is determined universally by local metric data. The supplied text does not state whether the conjecture has been proved or disproved.