Reshetikhin–Turaev comparison conjecture for modular tensor categories

Let $\mathcal{C}$ be a modular tensor category. Let $(\mathcal{A},q)$ be an isomorphism between $\mathrm{Vect}$ and $\mathcal{C}$ in ${\rm Braid}^{\otimes}$, equivalently an equivalence

$$q\colon\mathcal{C}\to\mathcal{Z}(\mathcal{A}).$$

The boundary condition $(\mathcal{A},q)$ determines an anomalous fully extended 3-dimensional theory $Z^{(\mathcal{A},q)}$, while $Z^{\rm RT}_{\mathcal{C}}$ denotes the anomalous 3-dimensional Reshetikhin–Turaev theory associated with $\mathcal{C}$. Reshetikhin–Turaev comparison conjecture. Any such $(\mathcal{A},q)$ induces a natural equivalence

$$Z^{\rm RT}_{\mathcal{C}}\simeq\Omega({Z^{(\mathcal{A},q)}}).$$

The claim proposes that the Reshetikhin–Turaev theory is obtained by applying the loop operator to the anomalous theory arising from the corresponding boundary condition. The supplied text presents this as a tempting conjecture and gives no resolution.