Ayoub's Hodge realisation conjecture for the limit mixed motive

Let $\mathcal{Z}^{\vee}\in\textbf{DM}_{\text{gm}}(\mathbb{Q},\mathbb{Q})$ be the dual limit mixed motive, and let $\mathbf{Z}^{\bullet}$ be the complex in $D^b(\textbf{MHS}_{\mathbb{Q}})$ whose cohomology computes the limit mixed Hodge structures. Ayoub's Hodge realisation conjecture. The Hodge realisation of $\mathcal{Z}^{\vee}$ is isomorphic to $\mathbf{Z}^{\bullet}$, equivalently, it is the complex in $D^b(\textbf{MHS}_{\mathbb{Q}})$ that computes the limit mixed Hodge structure. The statement is attributed in the source to Ayoub and is presented as conjectural.