  A hypersymplectic structure on a 4-manifold $X$ is a triple
$\underline{\omega}$ of symplectic forms which at every point span a maximal
positive-definite subspace of $\Lambda^2$ for the wedge product. This article
is motivated by a conjecture of Donaldson: when $X$ is compact
$\underline{\omega}$ can be deformed through cohomologous hypersymplectic
structures to a hyperk\"ahler triple. We approach this via a link with
$G_2$-geometry. A hypersymplectic structure $\underline\omega$ on a compact
manifold $X$ defines a natural $G_2$-structure $\phi$ on $X \times
\mathbb{T}^3$ which has vanishing torsion precisely when $\underline{\omega}$
is a hyperk\"ahler triple. We study the $G_2$-Laplacian flow starting from
$\phi$, which we interpret as a flow of hypersymplectic structures. Our main
result is that the flow extends as long as the scalar curvature of the
corresponding $G_2$-structure remains bounded. An application of our result is
a lower bound for the maximal existence time of the flow, in terms of weak
bounds on the initial data (and with no assumption that scalar curvature is
bounded along the flow).
