The Conforti--Cornuéjols packing problem

Let $\mathcal C$ be a clutter. Its packing property means that every minor $\mathcal C'$ satisfies the König property, namely $\alpha_0(\mathcal C')=\beta_1(\mathcal C')$. The max-flow min-cut property is the standard corresponding flow-cut property for a clutter.

Conforti--Cornuéjols packing problem. A clutter $\mathcal C$ has the packing property if and only if it has the max-flow min-cut property.

The text states that the converse of the known implication from max-flow min-cut to packing is an unsolved conjecture.