The rational orthogonal realization conjecture for rational zero-patterns

Let $X$ be an $m\times n$ matrix over a field. Its zero-pattern (or support) is the $(0,1)$-matrix $\underline{X}$ defined by

$$\underline{X}_{i,j}=\begin{cases}1,&X_{i,j}\neq0,\\0,&X_{i,j}=0.\end{cases}$$

Let $O(n)$ denote the real orthogonal $n\times n$ matrices, and let $O_n(\mathbb{Q})$ denote the orthogonal $n\times n$ matrices with rational entries. The rational orthogonal realization conjecture. For any $Y\in O(n)$ there exists $Z\in O_n(\mathbb{Q})$ such that $\underline{Y}=\underline{Z}$.

This is the rational analogue of the preceding unitary-versus-real zero-pattern conjecture: it asks whether every zero-pattern realized by a real orthogonal matrix is also realized by a rational orthogonal matrix. The abstract records verification of this claim for matrix size $n\leq5$, while no general resolution is supplied.