Critical threshold formula for higher critical number

Let $N$ be a simple binary matroid of critical number $c\geq 2$. A $k$-codimensional subspace of $N$ is obtained by intersecting its ground set, in a projective-geometry representation, with a flat of rank $r(N)-k$. Let $\mathcal{N}_0$, $\mathcal{N}_{1/4}$, and $\mathcal{N}_{1/2}$ be the subclasses of critical-number-two simple binary matroids defined in the source. Higher-critical-number threshold conjecture. The critical threshold of $N$ is

$$1-(1-\delta)2^{2-c},$$

where $\delta\in\{0,\tfrac14,\tfrac12\}$ is minimal such that $N|S\in\mathcal{N}_\delta$ for some $(c-2)$-codimensional subspace $S$ of $N$. This conjecture is stated to imply the simplified threshold-values conjecture and remains open in the source.