4 problems

Walsh's converse conjecture. For every matroid $K$ with a set $X$, there is a matroid $N$ on the circuits of $K/X$ such that

Walsh's lift conjecture. Every lift of $M$ is isomorphic to $M^N$ for some matroid $N$ on the circuits of $M$.

The explicit matroid construction conjecture. The collection $\mathcal{I}$ is the collection of independent sets of a matroid $N$ on $\mathcal{C}(M)$, and

Let $M$ be a matroid. A lift of $M$ is a matroid obtained from $M$ by increasing its rank while preserving the relevant lift relation. Let $\mathcal{C}(M)$ denote the set of circui…