Unbounded degree of irrationality for very general polarized K3 surfaces

Let $S_d$ be a very general polarized $K3$ surface with polarization $L_d$ such that $L_d^2=2d-2$, for each $d\in\mathbb N$. Unbounded irrationality conjecture.

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

The degree of irrationality measures how far a projective variety is from being rational, generalizing the gonality of curves. This conjecture, proposed in the cited work, remains open and motivates the study of birational invariants of hyper-Kähler manifolds.