The symmetric rational orthogonal zero-pattern conjecture

Let $U(n)$ denote the unitary $n\times n$ matrices and $O_n(\mathbb{Q})$ the orthogonal $n\times n$ matrices with rational entries. A matrix in $U(n)$ is hermitian when it equals its conjugate transpose; a matrix in $O_n(\mathbb{Q})$ is symmetric when it equals its transpose. For any matrix $X$, write $\underline{X}$ for its zero-pattern, defined entrywise by

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

The symmetric rational orthogonal zero-pattern conjecture. For any hermitian $X\in U(n)$ there exists a symmetric $Z\in O_n(\mathbb{Q})$ such that $\underline{X}=\underline{Z}$ and

$$\operatorname{Tr}(X)=\operatorname{Tr}(Z).$$

This is the combined symmetric analogue of the two preceding conjectures, adding equality of traces to equality of zero-patterns. The supplied text gives no proof or resolution of the assertion.