Conjecture on almost-tree, minimal and obstructing clutters

Let $d$ be a positive integer, and let $\mathscr{C}^{\rm a.tree}_d$, $\mathscr{C}^{\min}_d$, and $\mathscr{C}^{\rm obs}_d$ denote, respectively, the classes of $d$-uniform clutters that are almost trees, minimal to $d$-linearity, and obstructing to $d$-linearity. Equality conjecture.

$$\mathscr{C}^{\rm a.tree}_d=\mathscr{C}^{\min}_d=\mathscr{C}^{\rm obs}_d.$$

The paper has established the inclusions $\mathscr{C}^{\rm a.tree}_d\subseteq\mathscr{C}^{\min}_d\subseteq\mathscr{C}^{\rm obs}_d$; the conjecture asserts that both inclusions are equalities. Its status is not determined by the supplied text.