The algebraic-dimension conjecture for compact hyperkähler manifolds

Let $X$ be a non-projective compact hyperkähler manifold and let $L$ be a non-trivial nef line bundle. Let $a(X)$ be the transcendence degree of the field of meromorphic functions on $X$, and let $\kappa(X,L)$ be the Kodaira dimension of $L$. Algebraic-dimension conjecture for nef line bundles. Then

$$a(X)=\kappa(X,L).$$

The source notes that this follows when $L$ is semiample from results on hyperkähler fibrations, while the general statement remains conjectural.