Framed foam functoriality conjecture for annular Khovanov–Rozansky homology

Let $S$ be the surface supporting the tangled webs, let $\mathrm{S}\boldsymbol{\mathrm{Tan}}$ be the category of tangled webs and framed foams in four-dimensional space, and let $N\boldsymbol{\mathrm{Foam}}(\mathrm{S})$ be the corresponding foam category. Framed foam functoriality conjecture. The Khovanov–Rozansky functor extends to a functor from the category of tangled webs and framed foams in four-dimensional space to $K^b(N\boldsymbol{\mathrm{Foam}}(\mathrm{S}))$. A functorial extension would provide canonical cobordism maps for the framed version of the construction; the paper explicitly identifies this as an open question, since uniqueness or a distinguished lift to the framed setting is not known.