Braidedness conjecture for annular Khovanov–Rozansky functors

Let $\mathrm{A}\boldsymbol{\mathrm{Link}}^+$ be the category of consistently oriented annular links and cobordisms, and let $K^b(\mathrm{A}\boldsymbol{\mathrm{Foam}}^+)$ be the homotopy category of consistently oriented annular webs and foams. Braidedness conjecture. The annular Khovanov–Rozansky functor

$$\mathrm{A}\boldsymbol{\mathrm{Link}}^+\to K^b(\mathrm{A}\boldsymbol{\mathrm{Foam}}^+)$$

is braided. This would make the monoidal annular invariant compatible with the braiding on the entire consistently oriented annular link category. The paper proves the relevant behavior in special cases but states that the general result is not known because the required stronger functoriality has not been established.