Extends ParticipantInfo with cachedUser: User | null. In updateParticipants,
snapshots the previous participant list and carries forward the cached User
object when spaceStore member lookup fails (user navigated to a different
space). Local user falls back to authStore. homeUserId is now derived from
cachedUser when the direct member lookup misses. Updates EMPTY_PARTICIPANT
in PictureInPicture.tsx.