Classification conjecture for finite depth objects in bicommutant categories

Let $\mathcal{B}$ be a bicommutant category admitting an absorbing object whose endomorphism algebra is a hyperfinite $\mathrm{II}_\infty$ or $\mathrm{III}_1$ factor. Assume that

$$Z(\mathcal{B})\cong \operatorname{Hilb}(\mathcal{V})$$

for some unitary braided fusion category $\mathcal{V}$. Objects $X,Y\in\mathcal{B}$ are conjugate if there exists an invertible object $U\in\mathcal{B}$ such that $Y\cong UXU^{-1}$. Classification conjecture. There exists a natural bijective correspondence

$$\left\{\text{finite depth objects of }\mathcal{B}\right\}\!\left/\text{conj.}\right. \leftrightarrow \left\{\text{connected finite depth unitary anchored planar algebras in }\mathcal{V}\right\}\!\left/\text{iso.}\right..$$

This conjectures that the finite depth objects of such bicommutant categories are classified, up to conjugacy, by connected finite depth unitary anchored planar algebras in the associated unitary braided fusion category. The result extends the known classification for bimodule categories by planar algebras; the conjectural correspondence is the subject of the paper's broader classification program, while the supplied text does not establish its resolution.