Existence of a representing Borel measure for multiply connected cavities

Let $\omega=\cup_{k=1}^N\omega_k$ be a multiply connected set, where the sets $\omega_k$, for $k=1,\dots,N$, are nonintersecting simply connected open domains with $C^{1,1}$ boundaries $\gamma_k$, and $\overline{\omega}\subset\Omega$. Set $\gamma=\cup_{k=1}^N\gamma_k$. For $f\in\mathcal H^{\frac{1}{2}}(\gamma)$, let $F$ be a holomorphic function in $\omega$ whose trace on $\gamma$ is $f$. Representing-measure conjecture. There exists a Borel measure $\nu$ supported in $\omega$ such that

$$\langle f,Q^{1}_{\gamma}\rangle_{\frac12,\gamma}=\int_\omega F'\,\mathrm d\nu$$

for every such trace $f$ and holomorphic function $F$. This would extend the measure representation already proved for a simply connected cavity and for the case of two disks to arbitrary finite unions of nonintersecting simply connected cavities, underpinning the proposed shape-from-moments reconstruction method.