On a conjecture concerning dyadic oriented matroids (Q1288887)

Summary: A rational matrix is totally dyadic if all of its nonzero subdeterminants are in \(\{\pm 2^k\mid k\in\mathbb{Z}\}\). An oriented matroid is dyadic if it has a totally dyadic representation \(A\). A dyadic oriented matroid is dyadic of order \(k\) if it has a totally dyadic representation \(A\) with full row rank and with the property that for each pair of adjacent bases \(A_1\) and \(A_2\) \[ 2^{-k}\leq \Biggl|{\text{det}(A_1)\over \text{det}(A_2)}\Biggr|\leq 2^k. \] We present a counterexample to a conjecture on the relationship between the order of a dyadic oriented matroid and the ratio of agreement to disagreement in sign of its signed circuits and cocircuits (Conjecture 5.2, \textit{J. Lee} [J. Comb. Theory, Ser. B 50, No. 2, 265-287 (1990; Zbl 0657.05017)]).