Unique maximality conjecture for matroids avoiding dual symmetric-tensor circuits

Let $n,t$ be non-negative integers with $n\geq t+1$, and define

$${\cal X}=\{K_{a,b}:a,b\geq 1,\ a+b=t+2\}.$$

An ${\cal X}$-matroid on $K_n$ is a matroid in the family whose specified circuits are the complete bipartite graphs in ${\cal X}$.

Unique maximality conjecture. There is a unique maximal ${\cal X}$-matroid on $K_n$.

The preceding discussion says that the analogous family defined only by $K_{1,t+1}$ is not uniquely maximal when $t\geq 4$, motivating this conjecture; no resolution of the ${\cal X}$-matroid conjecture is supplied.