The GNRS conjecture on graph families and minor exclusion

For a graph family $\mathcal F$, let $c_1(\mathcal F)$ denote the supremum of the least distortion needed to embed its members into $L_1$.

GNRS conjecture. For every family of finite graphs $\mathcal F$, one has $c_1(\mathcal F)=O(1)$ if and only if $\mathcal F$ forbids some minor.

The conjecture would characterize graph families with uniformly bounded $L_1$-embedding distortion, equivalently those with uniformly bounded multi-commodity flow/cut gap. It generalizes the planar embedding problem and remains open.