Minimality conjecture for models of non-rational Hilbert modular surfaces

Let $N$ be an integer coprime to $6$, let $r$ be as in the construction, and let $Z^\circ_{N,r}$ and $W_{N,r}^{\mathsf{small}}$ be the models defined in the nonsingular-model and general-type constructions. Minimality conjecture. For all sufficiently large integers $N$ coprime to $6$, the surfaces $Z^\circ_{N,r}$ and $W_{N,r}^{\mathsf{small}}$ are minimal. The paper presents this as an open problem because minimal models are not known for the corresponding surfaces in general; it is the detailed formulation of the minimality expectation stated earlier.