The effective-line conjecture for hyper-Kähler manifolds

Let $X$ be a projective irreducible holomorphic symplectic manifold, and let $\widetilde{H}(X,\mathbb{Q})$ be its extended rational cohomology. A line $\ell$ in $\widetilde{H}(X,\mathbb{Q})$ is effective if it is of the form $\ell(F)$ for an object $F$ in $D^b(X)$ with a rank $1$ cohomological obstruction map. It is potentially effective if there exist a projective irreducible holomorphic symplectic manifold $Y$ and a derived parallel transport operator $\widetilde{\phi}:\widetilde{H}(X,\mathbb{Q})\to\widetilde{H}(Y,\mathbb{Q})$ such that $\widetilde{\phi}(\ell)$ is effective. Effective-line conjecture. A line $\ell$ in $\widetilde{H}(X,\mathbb{Q})$ is effective if it is potentially effective and is contained in $\widetilde{H}^{0,0}(X,\mathbb{Q})$. This is motivated by the modularity theorem for stable sheaves with a rank $1$ obstruction map and is intended to characterize effective lines through potential effectivity and Hodge type; the source gives no resolution.