Completion of the GIT proof

Proof of Theorem 57.1. The point $w=(V,C)$ is stable by Proposition 58.5. The smooth locus $W^\mathrm{sm}$ is irreducible, and its K-stable locus contains a nonempty open subset by the known general-member result and openness of K-stability. Choose an integral curve through $w$ whose generic point belongs to this open subset, normalize it, and restrict to a neighbourhood of a point above $w$. Its blow-ups form a smooth Fano family, and its punctured fibres may be taken K-stable.

Properness supplies a K-polystable filling after finite base change. Apply the relative reconstruction in [34, Proposition A.13] and Lemma 59.3. We conclude that $X=\mathop{\mathrm{Bl}}_CV$ is K-polystable. Finally, every automorphism of $X$ preserves its unique birational extremal contraction, so it descends to a projective automorphism of $V$ preserving $C$. Lemma 58.1 shows that this group is finite. K-polystability with a finite automorphism group is K-stability, completing the proof. ◻