Maximal determinant conjecture for 2-consecutive-ones matrices

Let $A\in\{0,1\}^{n\times n}$ be a matrix with the 2-consecutive ones property, meaning that the ones in each row occur in at most two blocks in some ordering of the columns. 2-consecutive-ones maximal determinant conjecture. Then

$$|\det(A)|\leq 4^{n/3}.$$

The paper gives constructions with determinant $4^{(n-3)/3}$ for positive integers $n$ divisible by $3$, and states that this construction is essentially optimal. The source provides no proof of the conjectured upper bound.