  We prove that for a locally stable family of klt singularities with constant
local volume, the ideal sequences of the minimizing valuations for the
normalized volume function form a family of ideals with flat cosupport, which
induces a degeneration to a locally stable family of K-semistable log Fano cone
singularities. Our proof is a family version of the method of C. Xu and Z.
Zhuang proving finite generation by Koll\'ar models and multiple degenerations.
