Geelen–Nelson critical-threshold conjecture for binary matroids

Let $N$ be a binary matroid with critical number $\bchi(N)=c$, and let $\btheta(N)$ denote its critical threshold. Geelen–Nelson's conjecture.

$$\theta(N)=1-i2^{-c},$$

where $i=2$ if and only if no $(c-1)$-codimensional subspace $S$ exists such that $S\cap N$ is a set of linearly independent vectors, $i=4$ if and only if $N$ is $c$-near-independent, and $i=3$ otherwise.

Geelen and Nelson prove that the displayed expression is a valid lower bound, while the general matching upper bound remains open; known results establish some special cases.