The hyperpencil conjecture for integral symplectic forms

Let $X$ be a closed $2n$-manifold and let $\omega$ be an integral symplectic form on $X$, meaning that its cohomology class lies in the image of $H^2(X;\mathbb{Z})\to H^2_{\operatorname{dR}}(X)$. Let $\Omega$ denote the map from deformation classes of hyperpencils on $X$ to isotopy classes of symplectic forms. Hyperpencil conjecture. For any sufficiently large integer $m$, the isotopy class of $m\omega$ lies in the image of $\Omega$.