A Hadamard matrix with growth factor greater than its order

It has long been conjectured that the growth factor of a completely pivoted Hadamard matrix is equal to its order. In this paper, we construct a row and column permutation of the Sylvester Hadamard matrix of order $4096$ that is completely pivoted and has growth factor $4608=(9/8)4096$, disproving the conjecture. This construction can be extended to give a completely pivoted Hadamard matrix of order $2^d$ with growth factor $(9/8)2^d$ for every integer $d\ge12$.