Stapleton's asymptotic conjecture for the degree of irrationality of K3 surfaces

Let $(S_d,L_d)$ denote a very general polarized $K3$ surface of degree $d$. Then there exist constants $C_1,C_2>0$ such that

$$C_1\sqrt{d}\leq \operatorname{irr}(S_d)\leq C_2\sqrt{d}.$$

Stapleton's asymptotic conjecture. The degree of irrationality of a very general polarized $K3$ surface grows on the order of $\sqrt d$. This refines the known upper bound and is consistent with the open expectation that the degree of irrationality is unbounded as the polarization degree grows.