Finiteness conjecture for density-critical matroids of prescribed density

For a matroid $M$, let $d(M)$ be its density, and call $M$ density-critical when every proper minor $N$ satisfies $d(N)<d(M)$. Finiteness conjecture. For every $t\geq 0$, there are finitely many density-critical matroids with density exactly $t$. This is presented as a weakening of the conjecture asserting finiteness of $t$-density-critical matroids, and remains open.