Geelen and Nelson's critical threshold conjecture for matroids

Let $N$ be a matroid, let $\chi(N)$ denote its critical number, and let its critical threshold be the infimum of the density thresholds forcing bounded critical number. Geelen and Nelson's conjecture. If $\chi(N)\geq 2$, then the critical threshold of $N$ is

$$1-i\cdot2^{-\chi(N)}$$

for some $i\in\\{2,3,4\\}$. The lower bound is known, while the matching upper-threshold description remains open in general.