Kinser class non-duality conjecture

For each integer $n>5$, let $\mathcal{K}_n$ be the class of matroids satisfying the $n$-th Kinser inequality, and let $\mathcal{K}_n^*$ denote its dual class. Kinser class non-duality conjecture.

$$\mathcal{K}_n\neq\mathcal{K}_n^*.$$

This asserts that, beyond the fifth Kinser class, the class defined by the $n$-th Kinser inequality is not closed under duality. The source gives no resolution.