From Lemma 2.1 there is an a priori volume bound $\text{Vol}(L)\leq C$. But if $P$ is in the support of $L$, then the regularity scale of $X$ near $P$ is bounded by

$$R^{n}\leq C\text{Vol}(L\cap B(R))\leq C,$$

whence $P$ must remain in a fixed compact subset. ∎