Non-vanishing conjecture for hyperkähler manifolds

Let $X$ be a not necessarily projective hyperkähler manifold, and let $L$ be a nef divisor on $X$ such that

$$q_X(L,L)=0,$$

where $q_X(\cdot,\cdot)$ denotes the Bogomolov--Beauville--Fujiki form on $X$.

Non-vanishing conjecture. One has $\kappa(L)\geq 0$.

This conjecture concerns non-vanishing for nef isotropic divisors on hyperkähler manifolds and is examined as a comparison for the paper's method. The supplied text does not state that it has been resolved.