  We show that the partial sums of the long Pl\"ucker relations for pairs of
weakly separated Pl\"ucker coordinates oscillate around $0$ on the totally
nonnegative part of the Grassmannian. Our result generalizes the classical
oscillating inequalities by Gantmacher--Krein (1941) and recent results on
totally nonnegative matrix inequalities by Fallat--Vishwakarma (2023). In fact
we obtain a characterization of weak separability, by showing that no other
pair of Pl\"ucker coordinates satisfies this property.
  Weakly separated sets were initially introduced by Leclerc and Zelevinsky and
are closely connected with the cluster algebra of the Grassmannian. Moreover,
our work connects several fundamental objects such as weak separability,
Temperley--Lieb immanants, and Pl\"ucker relations, and provides a very general
and natural class of additive determinantal inequalities on the totally
nonnegative part of the Grassmannian.
