The symmetry-defect braid-representation conjecture

Let $G$ act on a braided fusion category, let $X_g$ be a defect fixed by the $G$-action, and let $\rho_{X_g,n}$ be the associated projective representation of the braid group $\mathcal{B}_n$. For an irreducible projective representation $\pi$ of $G$, let $\rho_{(X_g,\pi),n}$ be the representation associated with the corresponding gauged anyon. Symmetry-defect representation conjecture. For every $n$ and $\pi$,

$$\rho_{X_g,n}\quad\text{and}\quad\rho_{(X_g,\pi),n}$$

are equivalent as projective representations. The source gives no resolution.