  The quantum max-flow min-cut conjecture relates the rank of a tensor network
to the minimum cut in the case that all tensors in the network are
identical\cite{mfmc1}. This conjecture was shown to be false in Ref.
\onlinecite{mfmc2} by an explicit counter-example. Here, we show that the
conjecture is almost true, in that the ratio of the quantum max-flow to the
quantum min-cut converges to $1$ as the dimension $N$ of the degrees of freedom
on the edges of the network tends to infinity. The proof is based on estimating
moments of the singular values of the network. We introduce a generalization of
"rainbow diagrams"\cite{rainbow} to tensor networks to estimate the dominant
diagrams. A direct comparison of second and fourth moments lower bounds the
ratio of the quantum max-flow to the quantum min-cut by a constant. To show the
tighter bound that the ratio tends to $1$, we consider higher moments. In
addition, we show that the limiting moments as $N \rightarrow \infty$ agree
with that in a different ensemble where tensors in the network are chosen
independently, this is used to show that the distributions of singular values
in the two different ensembles weakly converge to the same limiting
distribution. We present also a numerical study of one particular tensor
network, which shows a surprising dependence of the rank deficit on $N \mod 4$
and suggests further conjecture on the limiting behavior of the rank.
