Kalai–Whiteley maximality conjecture for hyperconnectivity matroids

Let $\mathcal{H}_d(n)$ be the generic hyperconnectivity matroid on the edges of the complete graph. Let $K_{d+2}$ be a complete graph on $d+2$ vertices and $K_{d+1,d+1}$ a complete bipartite graph with parts of size $d+1$.

Kalai–Whiteley maximality conjecture. The matroid $\mathcal{H}_d(n)$ is the freest matroid in which every $K_{d+2}$ and every $K_{d+1,d+1}$ are circuits.

This is known for $d=1$ and remains open for $d=2$; the source gives no resolution in higher dimensions. Partial results are attributed to Jackson and Tanigawa.