Let $N\in\mathbb N$, let $M\subset\mathbb C^N$ be a $C^\infty$ embedded generic CR submanifold of positive real codimension, and let $p\in M$. Let $T^{0,1}M\subset\mathbb C TM$ be the antiholomorphic CR bundle induced by the embedding. Assume that $M$ is minimal at $p$, meaning that the local CR orbit of $p$ generated by the real and imaginary parts of local smooth sections of $T^{0,1}M$ is an open neighbourhood of $p$ in $M$. Let $U_M\subset M$ be an open neighbourhood of $p$, and let $u:U_M\to\mathbb C$ be continuous and CR in the distributional tangential Cauchy-Riemann sense, meaning that $\bar L u=0$ in $\mathcal D'(U_M)$ for every smooth local section $\bar L$ of $T^{0,1}M$ on an open subset of $U_M$. Then there exist an open neighbourhood $V_M\subset U_M$ of $p$ in $M$, a smooth real vector subbundle $E\to V_M$ such that $T_q\mathbb C^N=T_qM\oplus E_q$ for every $q\in V_M$, an open conic subset $\Gamma\subset E$ such that each fibre $\Gamma_q:=\Gamma\cap E_q$ is a non-empty open convex cone in $E_q$, a number $\varepsilon>0$, a smooth tubular map $\Theta:\mathcal O_E\to\mathbb C^N$ from an open neighbourhood $\mathcal O_E\subset E$ of the zero section satisfying $\Theta(0_q)=q$ and whose differential $d\Theta_{0_q}:T_{0_q}E\to T_q\mathbb C^N$ restricts to the identity on $T_qM$ along the zero section and identifies the vertical tangent space $T_{0_q}(E_q)$ with $E_q$ under the splitting $T_q\mathbb C^N=T_qM\oplus E_q$, and an open wedge $\mathcal W\subset\mathbb C^N$ of the form $\mathcal W=\Theta(\{\eta\in\Gamma:0<|\eta|<\varepsilon\})$. There exists a [holomorphic function](/page/Holomorphic%20Function) $F:\mathcal W\to\mathbb C$ such that $F$ has boundary value $u|_{V_M}$ on the edge in the following sense: for every $q\in V_M$ and every open cone $\Gamma'_q\subset E_q$ whose closure in $E_q\setminus\{0\}$ is compactly contained in $\Gamma_q$, one has $\lim_{\eta\to 0,\ \eta\in\Gamma'_q}F(\Theta(\eta))=u(q)$.