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 $M(K_{m,n})$. For all $n \geq 3$, the $(3,n)$-mace is 4-connected and has at least $2n$ representations over any field $F$ of non-prime order $q \geq 9$. More generally, for $n \geq m$, the $(m,n)$-mace is vertically $(m+1)$-connected and has at least $2n$ inequivalent representations over any finite field of non-prime order $q\geq m^m$.