The algebraic-dimension values conjecture for compact hyperkähler manifolds

Let $X$ be a compact hyperkähler manifold of dimension $2n$, and let $a(X)$ denote its algebraic dimension. Algebraic-dimension values conjecture. Then

$$a(X)\in\{0,n,2n\}.$$

The source reports that this is known in dimension $4$ and under an appropriate minimal-model assumption in lower dimensions, but remains open in general.