The conical deficit global AdS$_3$ geometry is given by
$$ds^{2} = - \cosh^{2}\rho\ dt^{2} + d\rho^{2} + \sinh^{2}\rho\ d\varphi^{2}, \qquad 0 \leq \varphi < 2\pi(1-4Gm'),$$
where $0 \leq 1-4Gm' < 1$, and $0 \leq \rho < \infty$.
Does a finite boundary condition $0 \leq \rho < \rho_c$ change the inequality $0 \leq \varphi < 2\pi(1-4Gm')$ in any way?
My hunch is that there is no change as the $\phi$ coordinate and the $\rho$ coordinate are independent.
