Gap conjecture for determinants of Seidel tournament matrices

For even $n$, let $\mathcal{D}(n)$ denote the set of square roots of determinants of $n\times n$ Seidel tournament matrices. The preceding argument shows that, assuming the existence of a skew-conference matrix, a gap occurs near the largest possible element of $\mathcal{D}(n)$. Gap conjecture. There is a gap in $\mathcal{D}(n)$ for all even $n\geq 8$. The source notes that the argument does not establish this without assuming a skew-conference matrix and suspects that this assumption is unnecessary.