Quantitative separation conjecture for Aarnes and symplectic quasi-states

Let $M$ be a closed connected symplectic manifold with symplectic form $\omega$, and endow $M$ with a metric inducing its topology. Let ${\mathcal{S}}(M,\omega)$ be the collection of symplectic quasi-states on $(M,\omega)$, and let $\Delta(M)$ be the set of delta-measures on $M$. Then there exists a neighborhood ${\mathcal{U}}$ of $\Delta(M)$ in the space of quasi-states on $M$, and a constant $c\in(0,1]$, such that every Aarnes quasi-state $\zeta\in{\mathcal{U}}$ satisfies

$$W_1(\zeta,{\mathcal{S}}(M,\omega))\geq c\cdot W_1(\zeta,\Delta(M)).$$

This proposes a quantitative refinement of the result that, on a closed connected symplectic manifold with $H^1(M;{\mathbb{Z}})=0$ and dimension at least four, the intersection of the collections of Aarnes and symplectic quasi-states consists exactly of delta-measures. It asserts that near the delta-measures, an Aarnes quasi-state remains quantitatively separated from the symplectic quasi-states in proportion to its distance from the delta-measures.