Donald's deformation conjecture for compact hypersymplectic 4-manifolds

Let $X$ be a compact 4-manifold and let $\underline{\omega}=(\omega_1,\omega_2,\omega_3)$ be a hypersymplectic structure, meaning a triple of symplectic forms such that every non-zero linear combination is symplectic. Suppose that

$$\int \omega_i\wedge\omega_j=2\delta_{ij}.$$

Donald's conjecture. The structure $\underline{\omega}$ can be deformed through cohomologous hypersymplectic structures to the triple of Kähler forms arising from a hyperkähler metric on $X$. In particular, $X$ is diffeomorphic to $\mathbb{T}^4$ or a K3 surface. The paper proves this conjecture under the additional assumption that the initial structure is invariant under an effective $S^1$-action; without that assumption, the general conjecture remains open.