The associator scheme conjecture for Ran-space fiber functors

Let $\mathcal M^{B}_{un}(Ran(\mathbb A^1))$ and $\mathcal M^{dR}_{un}(Ran(\mathbb A^1))$ be the Betti and de Rham categories equipped with their natural tensor structures and fiber functors $\omega_{Ran}^{B}$ and $\omega_{Ran}^{dR}$. Let $RH$ denote the Riemann–Hilbert functor, and let $\operatorname{Ass}(\mathbb C)$ be the scheme of Drinfeld associators. Associator scheme conjecture. The scheme of tensor isomorphisms between the fiber functors is isomorphic to the associator scheme:

$$\operatorname{Isom}^{\otimes}(\omega_{Ran}^{dR},\omega_{Ran}^{B} RH)\cong \operatorname{Ass}(\mathbb C).$$

This identifies the choices of tensor-compatible comparison between the de Rham and Betti fiber functors with Drinfeld associators. The supplied text states the isomorphism but gives no indication that it has been proved or disproved.