Extension of CR functions from boundaries in ${\mathbb{C}}^n\times \mathbb{R}$

Let $\mathrm{\Omega }\subset {\mathbb{C}}^n\times \mathbb{R}$ be a bounded domain with smooth boundary such that $\mathrm{\partial }\mathrm{\Omega }$ has only nondegenerate elliptic CR singularities, and let $f\text{\hspace{0.17em}⁣}:\mathrm{\partial }\mathrm{\Omega }\to \mathbb{C}$ be a smooth function that is CR at CR points of $\mathrm{\partial }\mathrm{\Omega }$ (when $n=1$ we require separate holomorphic extensions for each real parameter). Then $f$ extends to a smooth CR function on $\bar{\mathrm{\Omega }}$, that is, an analogue of Hartogs-Bochner holds. In addition, if $f$ and $\mathrm{\partial }\mathrm{\Omega }$ are real-analytic, then $f$ is the restriction of a function that is holomorphic on a neighborhood of $\bar{\mathrm{\Omega }}$ in ${\mathbb{C}}^{n+1}$. An immediate application is a (possibly singular) solution of the Levi-flat Plateau problem for codimension 2 submanifolds that are CR images of $\mathrm{\partial }\mathrm{\Omega }$ as above. The extension also holds locally near nondegenerate, holomorphically flat, elliptic CR singularities.