Descent for the cotangent complex along faithfully flat SCRs

By Theorem 3.1 of Bhatt-Morrow-Scholze II (https://arxiv.org/pdf/1802.03261.pdf), we know that for $ R$ a commutative ring, $ \wedge^{i}L_{(-)/R}$ satisfies descent for faithfully flat maps $ A \rightarrow B$ of (ordinary) $ R$ -algebras. Can this descent statement be promoted to, say, faithfully flat maps of simplicial commutative $ R$ -algebras (where $ R$ is either discrete or a simplicial commutative ring)? I am primarily interested in the case where $ i=1$ , as the proof of this claim along with an induction argument using a filtration, can probably give the $ i>1$ case.