The meromorphic-degree conjecture for hyper-Kähler manifolds

Let $N$ be a hyper-Kähler manifold of complex dimension $2n$, and suppose its Néron–Severi group has rank one and is generated by an isotropic element of $H^{2}(N,\mathbb{Z})$. The meromorphic-degree conjecture. The field of meromorphic functions on $N$ has transcendence degree $n=\frac{1}{2}\dim_{\mathbb{C}}N$. The statement is introduced as a conjecture motivated by holomorphic Morse techniques; the paper gives no resolution.