Estimate for symmetric functions in the range of the spectral projection complement

Let $\Lambda$ be the Dirichlet-to-Neumann operator, let $P_\tau$ denote the spectral projection associated with the threshold $\tau$, and let $J$ be the operator mapping $v$ to $w=Jv$. A function $v$ is symmetric when it belongs to the symmetric subspace, and set $w=Jv$. Estimate for symmetric functions. For any $\tau>1$ and any $w=Jv$ with symmetric $v\in \operatorname{Ran}(I-P_\tau)$, estimate $\textup{(A-24)}$ holds. The estimate is identified in the source as a needed step for extending the sharp spectral asymptotics to operators in domains with inner edges; the supplied text does not state whether it has been proved or remains open.