Kinser-minus separation conjecture

For an integer $n>5$, let ${\mathrm{Kin}(n)}^-$ denote the matroid referred to in the source as ${\mathrm{Kin}(n)}^-$. Let $\mathcal{K}_n$ be the class of matroids satisfying the $n$-th Kinser inequality, and let $\mathcal{K}_n^*$ be its dual class. Kinser-minus separation conjecture.

$${\mathrm{Kin}(n)}^-\in\mathcal{K}_n^* - \mathcal{K}_n.$$

The source presents this as a strengthening of the conjecture that $\mathcal{K}_n$ is not dual closed, motivated by the difficulty of the fifth case. The notation ${\mathrm{Kin}(n)}^-$ is not defined in the supplied context.