Finiteness conjecture for strictly density-critical matroids

For a matroid $M$, let $\d(M)$ denote its density, and call $M$ strictly $t$-density-critical when $d(M)>t$ while every proper minor $N$ of $M$ satisfies $d(N)\leq t$. Finiteness conjecture. For every $t\geq 0$, there are finitely many strictly $t$-density-critical matroids. The case $t=2$ is known from the explicit classification of the ten strictly $2$-density-critical matroids; the assertion for all $t\geq 0$ remains open.