The extended future tube conjecture for SO(1,n)

Let C be the open upper light cone in $\mathbb R^{1+n}$ with respect to the Lorentz product. The connected linear Lorentz group SO$_\mathbb R(1,n)^0$ acts on C and therefore diagonally on the N-fold product $T^N$ where $T = \mathbb R^{1+n} + iC \subset \mathbb C^{1+n}$. We prove that the extended future tube SO$_\mathbb C(1,n) \cdot T^N$ is a domain of holomorphy.