Rationality conjecture for symmetric Hilbert modular surfaces

Let $Z^{\mathrm{sym}}_{N,r}$ denote the symmetric Hilbert modular surface associated with an $(N,r)$-congruence, and suppose $(N,r)$ is one of the pairs in part (i) of the paper's classification theorem. Rationality conjecture. For each such $(N,r)$, the surface $Z^{\mathrm{sym}}_{N,r}$ is rational over $\mathbb{Q}$, equivalently birational over $\mathbb{Q}$ to $\mathbb{A}^2$. This conjecture extends the explicitly computed rational cases and concerns rationality over $\mathbb{Q}$ rather than only geometric rationality.