Hansen and Zwick's Fibonacci conjecture for Order-Regular matrices

Let an Order-Regular matrix be a binary matrix satisfying the Order-Regularity constraints arising from Policy Iteration on cube acyclic Unique Sink Orientations. For an $n$-column Order-Regular matrix, let $F_k$ denote the $k$th Fibonacci number. Hansen and Zwick's Fibonacci conjecture. The maximum number of rows of an $n$-column Order-Regular matrix is given by $F_{n+2}$, the $(n+2)$nd Fibonacci number.

Hansen and Zwick proposed this conjecture after exhaustive searches for matrices with up to six columns, whose maximum row counts were $2,3,5,8,13,21$. The paper disproves the conjecture by constructing Order-Regular matrices exceeding the corresponding Fibonacci bounds.