Kinser hierarchy dual-containment conjecture

For each integer $n>4$, 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 hierarchy dual-containment conjecture.

$$\mathcal{K}_{n+1}^*\subseteq\mathcal{K}_n.$$

This conjecture describes how the dual of each Kinser class is expected to lie inside the preceding class. The source gives no resolution.