The congruence twin conjecture for odd-dimensional integral vectors

Let $n\geq 5$ be an odd integer, let $d$ be an integer, and let a vector have length $\sqrt{d}$. Two vectors are twins when they are orthogonal and have the same norm.

Congruence twin conjecture. Suppose that the dimension $n\geq 5$ is odd and $d\not\equiv n\pmod{8}$. Then every vector of length $\sqrt{d}$ has a twin.

This is explicitly presented as a weaker conjecture than the assertion that every non-odd vector in odd dimension at least five has a twin: every odd vector of length $\sqrt{d}$ satisfies $d\equiv n\pmod{8}$.