Let lambda, mu, lambda', mu' be partitions. The conjecture of Lam, Postnikov and Pylyavskyy states that, if (lambda, mu, lambda', mu') obey certain natural inequalities, then s_{lambda'} s_{mu'} - s_{lambda} s_{mu} is Schur nonnegative. We prove this conjecture. Our proof is based on two key ideas. First, we introduce a new combinatorial model for Littlewood-Richardson coefficients which we name ``skeps", which are similar to but distinct from Knutson and Tao's hives. Second, we use tools from Murota's theory of L-convexity to prove an L-log-concavity theorem for skeps.