On universal quadratic inequalities for minors of TNN matrices

For positive integers $n,n^{\mathrm{\prime }}$, we give a combinatorial characterization for the set of quadratic inequalities on minors that are valid for all $n\times n^{\mathrm{\prime }}$ 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 Lindström’s construction.