Quasi-geometricity conjecture for connections from braided fusion categories

Let $\mathcal C$ be a braided fusion category over $\Bbb C$. For objects $X_1,\ldots,X_n,Y\in\mathcal C$, the space $\operatorname{Hom}(Y,X_1\otimes\cdots\otimes X_n)$ carries an action of the pure braid group $PB_n$, defining a Betti local system $\mathcal L_{X_1,\ldots,X_n,Y}$ on the configuration space

$$X_n=\Bbb C^n\setminus\{\text{diagonals}\}.$$

Let $\nabla_{X_1,\ldots,X_n,Y}$ be the unique, up to isomorphism, regular-singular flat connection on $X_n$ whose monodromy is this local system. A connection is quasi-geometric when it is defined over $\overline{\Bbb Q}$ in both its de Rham and Betti realizations. Quasi-geometricity conjecture. The connection $\nabla_{X_1,\ldots,X_n,Y}$ is defined over $\overline{\Bbb Q}$; equivalently, it is a quasi-geometric connection. The source notes that braided fusion categories are defined over $\overline{\Bbb Q}$ by Ocneanu rigidity, while the conjectural assertion concerns the resulting Riemann-Hilbert connections.