  Recently, much attention has been given to various inequalities among
partition functions. For example, Nicolas, {and later DeSavlvo--Pak,} proved
that $p(n)$ is eventually log-concave, and Ji--Zang showed that the cranks are
eventually unimodal. This has led to a flurry of recent activity generalizing
such results in different directions. At the same time, Stanton recently made
deep conjectures on the positivity of certain polynomials associated to ranks
and cranks of partitions, with the ultimate goal of pointing the way to
``deeper'' structure refining ranks and cranks. These have been shown to be
robust in recent works, which have identified further infinite families of such
conjectures in the case of colored partitions. In this paper, we employ the
Circle Method to prove unimodality for ranks. As a corollary, we prove
Stanton's original conjecture. This points to future study of the positive,
integral coefficients Stanton conjectured to exist, hinting at new
combinatorial structure yet to be uncovered.
