Continuation of separately analytic functions defined on part of the domain boundary

Let $D\subset\mathbb C^n$ be a domain with smooth boundary $\partial D$, let $E\subset\partial D$ be a subset of positive Lebesgue measure $\operatorname{mes}(E)>0$, and let $F\subset G$ be a nonpluripolar compact set in a strongly pseudoconvex domain $G\subset\mathbb C^m$. We prove that, under an additional condition, each function separately analytic on the set $X=(D\times F)\cup(E\times G)$ has a holomorphic contination to the domain $\widehat X=\{(z,w)\in D\times G:\omega_{\mathrm{in}}^*(z,E,D)+\omega^*(w,F,G)<1\}$, where $\omega^*$ is the $P$-measure and $\omega^*_{\mathrm{in}}$ is the interior $P$-measure.