Seymour's quarter-integral packing conjecture for ideal clutters

Let $\mathcal{C}$ be a clutter over ground set $V$. A packing is a nonnegative vector $y\in\mathbb{R}^{\mathcal{C}}_+$ satisfying

$$\sum_{C\in\mathcal{C}:v\in C}y_C\leq 1\qquad\text{for every }v\in V.$$

Its value is $\mathbf{1}^{\mathsf T}y$, and $\tau(\mathcal{C})$ denotes the covering number of $\mathcal{C}$. The packing is $\frac14$-integral if every nonzero coordinate is at least $\frac14$ and is an integral multiple of $\frac14$.

Seymour's quarter-integral packing conjecture. Every ideal clutter $\mathcal{C}$ has a $\frac14$-integral packing of value $\tau(\mathcal{C})$.

This conjecture is described as notoriously difficult and remains open even for binary clutters and for the clutter of postman sets of a graph. The paper derives consequences of it for wise-intersecting ideal clutters.