Donaldson's conjecture on hypersymplectic structures and hyperkähler structures

Let $M$ be a compact oriented $4$-manifold, and let $\underline{\omega}$ be a hypersymplectic structure on $M$. A one-parameter family of diffeomorphisms $F_t:M\to M$ connected to the identity is required to take $\underline{\omega}(0)=\underline{\omega}$ to a hyperkähler structure $\underline{\omega}(1)=F_1^*\underline{\omega}$, with

$$\underline{\omega}(t)=F_t^*\underline{\omega}$$

remaining in the same cohomology class for $0\leq t\leq 1$.

Donaldson's conjecture. For every such $M$ and $\underline{\omega}$, there exists a family with these properties.

The conjecture asks whether every compact hypersymplectic $4$-manifold can be carried, within its cohomology class and up to isotopy, to a hyperkähler structure. The paper studies the associated hypersymplectic flow, but does not establish the conjecture.