On inequivalent representations of matroids over non-prime fields

For each finite field F of prime order there is a constant c such that every 4-connected matroid has at most c inequivalent representations over F. We had hoped that this would extend to all finite fields, however, it was not to be. The (m,n)-mace is the matroid obtained by adding a point freely to M(Km,n). For all n ≥ 3, the (3, n)-mace is 4-connected and has at least 2 representations over any field F of non-prime order q ≥ 9. More generally, for n ≥ m, the (m,n)-mace is vertically (m + 1)-connected and has at least 2 inequivalent representations over any finite field of non-prime order q ≥ m.