The degree of irrationality of very general polarized K3 surfaces

Let $\{(S_d,L_d)\}_{d\in\mathbb N}$ be very general polarized $K3$ surfaces with $L_d^2=2d-2$. Bastianelli–De Poi–Ein–Lazarsfeld–Ullery conjecture.

$$\limsup_{d\to\infty}\operatorname{Irr}(S_d)=+\infty.$$

This predicts that the degrees of irrationality of very general polarized K3 surfaces are unbounded; the source provides no resolution.