Existence conjecture for symmetric $[\alpha]$-Hadamard matrices

A symmetric $[\alpha]$-Hadamard matrix of order $\zeta$ is a square $\zeta\times\zeta$ matrix with entries in $\{1,-1\}$ that is symmetric and whose distinct-row inner products have absolute value at most $\alpha$. Since ordinary Hadamard matrices are known not to exist in orders not divisible by $4$, the paper introduces these generalized matrices.

Existence conjecture for symmetric $[\alpha]$-Hadamard matrices. For every $\alpha\geq 3$, there exist symmetric $[\alpha]$-Hadamard matrices of all orders.

If true, this would extend the constructions based on Hadamard matrices to every order and would make the paper's multipartite Ramsey-number results applicable for all orders $\zeta$.