Holomorphic extension of functions along finite families of complex straight lines in an $n$-circular domain

We consider the continuous functions on the boundary of a bounded $n$-circular domain $D$ in $\mathbb C^n$, $n>1$, which admit one-dimensional holomorphic extension along a family of complex straight lines passing through finitely many points of $D$. The question is addressed of the existence of a holomorphic extension of these functions to $D$.