Exponential density conjecture for matroids excluding an induced $I_t$-restriction

Let $t\geq 2$ and $r\geq t$ be integers. A matroid $M=(E,G)$ is full-rank when $G=\operatorname{cl}(E)$; it is $r$-dimensional when $\dim(G)=r$. Let $I_t$ denote the $t$-dimensional matroid being excluded, and let $s$ be the remainder of $r$ on division by $t-1$. A flat is a projective subgeometry of $G$.

Exponential density conjecture. If $M=(E,G)$ is a full-rank, $r$-dimensional matroid with no induced $I_t$-restriction, then

$$|E|\geq s2^{\left\lceil r/(t-1)\right\rceil}+(t-1-s)2^{\left\lfloor r/(t-1)\right\rfloor}-(t-1).$$

Equality holds precisely when $E$ is the disjoint union of flats $F_1,\dotsc,F_{t-1}$ whose dimensions sum to $r$, with $|\dim(F_i)-\dim(F_j)|\leq 1$ for all $i,j$.

This generalises the proved density theorem for claw-free matroids. The paper says that even the case $t=3$ is difficult and that the conjecture is not known in the supplied text.