Walker's Drinfeld-center conjecture for rectangular and annular relation categories

Let $\Lambda$ be a $\mathbb{C}$-linear category. Let $\Lambda^{\mathcal R}$ and $\Lambda^{\mathcal A}$ be, respectively, the rectangular and annular versions of the corresponding locally defined picture or relation category. Write $\mathcal D$ for the Drinfeld center, or quantum double, construction, and $\operatorname{Rep}$ for the representation category. Walker's conjecture. One has

$$\mathcal D\bigl(\operatorname{Rep}(\Lambda^{\mathcal R})\bigr)=\operatorname{Rep}(\Lambda^{\mathcal A}),$$

that is, the Drinfeld center of the representation category of the rectangular picture category is isomorphic to the representation category of the corresponding annular category. This is presented as Walker's conjecture concerning the connection between affine representations and the quantum double in the group planar algebra case; the supplied text gives no resolution status.