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.