Finiteness conjecture for polarised hyperkähler manifolds with non-pseudoeffective twisted cotangent bundles

Let $2n$ be an even natural number. Let $(X,H)$ be a polarised hyperkähler manifold, where $X$ has dimension $2n$ and $H$ is an ample Cartier divisor. Write

$$\pi:\mathbb{P}(\Omega_X)\rightarrow X$$

for the projectivised cotangent bundle, and let $\zeta$ be its tautological class. Finiteness conjecture. For fixed $2n$, there exist only finitely many deformation families of such polarised hyperkähler manifolds $(X,H)$ for which $\zeta+\pi^*H$ is not pseudoeffective. The conjecture proposes a finiteness statement for the failure of pseudoeffectivity of twisted cotangent bundles in fixed dimension; the supplied text gives no resolution or further evidence, so its status remains open.