  We prove the folklore endpoint multilinear $k_j$-plane conjecture originated
from the paper \cite{bennett2006multilinear} of Bennett, Carbery and Tao. Along
the way we prove a more general result, namely the endpoint multilinear
$k_j$-variety theorem. Finally, we generalize our results to the endpoint
perturbed Beascamp-Lieb inequality using techniques in this paper.
