Horizontal-subspace conjecture for geometric Hodge structures

Let $H$ be a geometric Hodge structure, and let $H^{\nabla}$ denote its horizontal subspace. Horizontal-subspace conjecture. For any geometric Hodge structure $H$, the horizontal subspace $H^{\nabla}$ is a Hodge substructure isomorphic to a power of $\mathbb{Q}(0)$. The preceding proposition in the source identifies the horizontal part after extension of scalars with a Hodge substructure having horizontal Hodge filtration, but the source gives no evidence that this stronger assertion has been proved or disproved.