Gupta–Newman–Rabinovich–Sinclair conjecture on minor-closed flow gaps

A family of finite graphs is considered, and $c_1(\mathcal F)$ denotes the supremum of $c_1(G)$ over graphs $G$ in $\mathcal F$, where $c_1(G)$ is the multi-commodity max-flow/min-cut gap for $G$. A graph family forbids a minor if there is a graph that is not a minor of any member of the family.

Gupta–Newman–Rabinovich–Sinclair conjecture. For every family of finite graphs $\mathcal F$, one has

$$c_1(\mathcal F)<\infty$$

if and only if $\mathcal F$ forbids some minor.

This conjecture characterizes precisely the graph families with uniformly bounded multi-commodity max-flow/min-cut gap, equivalently bounded distortion into distributions over trees. The supplied text does not state whether it has been resolved.