Elliptic-fibration conjecture for symmetric Hilbert modular surfaces

Let $Z^{\mathrm{sym}}_{N,r}$ be the symmetric Hilbert modular surface, and consider each $(N,r)$ in parts (ii)–(iii) of the paper's classification theorem. A Jacobian elliptic fibration is an elliptic fibration equipped with a section. Elliptic-fibration conjecture. For every such $(N,r)$, there exists a Jacobian elliptic fibration

$$Z^{\mathrm{sym}}_{N,r}\dashrightarrow\mathbb{P}^1$$

defined over $\mathbb{Q}$; moreover, one can choose such a fibration with a $\mathbb{Q}$-rational section of infinite order. The conjecture asks for arithmetic models and sections for the elliptic cases in the classification.