I am confused with one aspect regarding uniqueness of solutions to the Klein-Gordon equation with specific boundary conditions in the Lorentzian manifold ${\rm AdS}_{d+1}$ and its Riemannian counterpart ${\rm EAdS}_{d+1}$. More specifically, in this paper the author claims in page 4 that, in Euclidean Anti-de Sitter space, given the Klein-Gordon action $$I[\phi]=\dfrac{1}{2}\int d^{d+1}x\sqrt{g}(|d\phi|^2+m^2\phi^2)$$ and given some function $\phi_0(\mathbf{x})$ there is one unique solution $\phi(z,\mathbf{x})$ which is regular in the interior and that behaves like $$\phi(z,\mathbf{x})=z^{2h_-}\phi_0(\mathbf{x})$$ as $z\to 0$, where $(z,\mathbf{x})$ are Poincaré coordinates and where $h_\pm = \frac{d}{2}\pm\sqrt{\frac{d^2}{4}+m^2}$. This unique solution can be written in terms of a Green's function, the bulk-to-boundary propagator:
$$\phi(z,\mathbf{x})=c\int d^d\mathbf{x}'\dfrac{z^{2h_+}}{(z^2+|\mathbf{x}-\mathbf{x}'|^2)^{2h_+}}\phi_0(\mathbf{x}').$$
Now in the Lorentzian case, again working in the Poincaré patch, the authors say in page 6 that a general nonsingular solution in the interior which approaches $z^{2h_-}\phi_0(\mathbf{x})$ as $z\to 0$ is given by $$\phi(z,\mathbf{x})=\phi_n(z,\mathbf{x})+c \int d^d\mathbf{x}' \dfrac{z^{2h_+}}{(z^2+|\mathbf{x}-\mathbf{x}'|^2)}\phi_0(\mathbf{x}'),$$ where $\phi_n(z,\mathbf{x})$ is a normalizable solution with $\phi_n(z,\mathbf{x})=z^{2h_+}\widetilde{\phi}_n(\mathbf{x})$ as $z\to 0$ and with the distance in the denominator being $|\mathbf{x}-\mathbf{x}'|^2=-(t-t')^2+\sum_{i=1}^{d-1}(x_i-x_i')^2$.
In any way, now the boundary condition does not fully determine the solution. In fact this $\phi_n(z,\mathbf{x})$ seems to remain unconstrained other than being a regular solution to the Klein-Gordon differential equation.
Question: Why is this happening here? Why the equations plus boundary conditions determine a unique solution (as seems natural) in the Riemannian signature but does not determine a unique solution in the Lorentzian signature? In other words, why in the Riemannian signature given $\phi_0(\mathbf{x})$ and the boundary condition one fully determines the solution while in the Lorentzian case $\phi_0(\mathbf{x})$ plus the differential equation does not fully determine the solution?