Lim-perfectoid purity implies perfectoid purity and injectivity

Let $R$ be a ring for which the notions of lim-perfectoid pure, perfectoid pure, lim-perfectoid injective, and perfectoid injective are defined. The lim-perfectoid comparison conjecture. If $R$ is lim-perfectoid pure, then $R$ is perfectoid pure; respectively, if $R$ is lim-perfectoid injective, then $R$ is perfectoid injective. The preceding equivalence for local complete intersections motivates asking whether the LCI hypothesis can be weakened; the source gives no proof of this implication.