The disjoint curves conjecture for Kähler manifolds and orbifolds

Let $X$ be a Kähler manifold or a Kähler orbifold with first Betti number $b_1=0$ and second Betti number $b_2\geq 2$. Suppose that $X$ contains disjoint complex curves spanning $H_2(X,\mathbb{Q})$, all of the same genus $g\geq 1$. Disjoint curves conjecture. There does not exist such an $X$. This conjecture constrains the topology of Kähler manifolds and orbifolds containing disjoint complex curves that span their second homology. The cited result gives a bound in a related setting, but the nonexistence statement itself remains open here.