Existence of Hadamard matrices for all multiples of four

Determine whether for every positive integer n divisible by 4, there exists an n×n Hadamard matrix H with entries ±1 satisfying HH^T = nI_n. Ascertain existence in particular for the smallest outstanding order 768.

Background

Hadamard matrices H_n ∈ {±1}^{n×n} satisfy HH^T = nI_n and necessarily exist only for n = 1, 2 or n multiple of 4. Despite extensive constructions (e.g., Sylvester, Paley, Williamson), existence is not known for all multiples of 4. The smallest open case currently cited is order 768, following the resolution of order 764.

This paper discusses butterfly Hadamard matrices as a rich subclass that can be explicitly constructed and counted, but it does not resolve the general existence question for all multiples of 4.