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

Suppose that $D\subset\mathbb C^n$ is a domain with smooth boundary $\partial D$, $E\subset\partial D$ is a boundary subset of positive Lebesgue measure $\operatorname{mes}(E)>0$, and $F\subset G$ is a nonpluripolar compact set in a strongly pseudoconvex domain $G\subset\mathbb C^m$. We prove that, under some additional conditions, each function separately analytic on the set $X=(D\times F)\cup(E\times G)$ can be holomorphically continued into the domain $\widehat X=\{(z,w)\in D\times G:\omega_{\textup{in}}^*(z,E,D)+\omega^*(w,F,G)<1\}$, where $\omega^*$ is the $P$-measure and $\omega^*_{\textup{in}}$ is the inner $P$-measure.