Critical thresholds for critical-number-two binary matroids

Let $\mathcal{N}$ be the class of simple binary matroids of critical number $2$. For $\delta\in\{0,\tfrac14,\tfrac12\}$, let $\mathcal{N}_\delta$ be the three subclasses defined in the source according to the existence and properties of a $1$-codimensional independent subspace. Critical-number-two threshold conjecture. For each $\delta\in\{0,\tfrac14,\tfrac12\}$, every matroid in $\mathcal{N}_\delta$ has critical threshold $\delta$. This is the first case of the paper's proposed classification of critical thresholds; it is not resolved in the source.