By Proposition A.3, $V$ is proper if and only if the map $\tilde{X_{x}}\rightarrow\boldsymbol{P}_{\widetilde{\mathscr{H}(x)}/\tilde{K}}$ is bijective which by [Tem04, Theorem 5.2] is equivalent to the map $X_{x}\rightarrow\mathscr{M}(K)$ being closed. But this is equivalent to $x$ being an interior point of $X$. ∎