The odd-dimensional twin conjecture for integral vectors

Let $n\geq 5$ be an odd integer. An integral vector is odd if every component is odd; a vector is non-odd if it is not odd. Two vectors are twins when they are orthogonal and have the same norm.

Odd-dimensional twin conjecture. Suppose that the dimension $n\geq 5$ is odd. Then every non-odd vector has a twin.

Odd vectors cannot have twins in odd dimensions by reducing their components modulo $2$, so the conjecture identifies the non-odd vectors as the cases where a twin should always exist. It concerns the construction and extension of orthogonal integral vectors in higher-dimensional spaces.