On universal quadratic inequalities for minors of TNN matrices

In this note, for positive integers $n,n'$, we give a geometric-combinatorial characterization for the set of quadratic inequalities on minors that are valid for all $n\times n'$ totally nonnegative matrices. This is obtained as a consequence from our earlier results on ``stable'' quadratic identities on minors of matrices generated by flows in planar graphs via Lindstr\"om's construction.