  We study the $S_n$-equivariant log-concavity of the cohomology of flag
varieties, also known as the coinvariant ring of $S_n$. Using the theory of
representation stability, we give computer-assisted proofs of the equivariant
log-concavity in low degrees and high degrees and conjecture that it holds for
all degrees. Furthermore, we make a stronger unimodal conjecture which implies
the equivariant log-concavity.
