The annular web evaluation equivalence conjecture

Let $\mathcal{U}_Q(\mathfrak{gl}_m)^{0\leq}$ be the quotient of categorified quantum $\mathfrak{gl}_m$, and let $n\operatorname{Foam}$ denote the corresponding foam category for annular webs. Write $\operatorname{hTr}$ for the horizontal trace and $\widetilde{\operatorname{vTr}}$ for the modified vertical trace.

Annular web evaluation conjecture. There is an equivalence of categories

$$\operatorname{hTr}(\mathcal{U}_Q(\mathfrak{gl}_m)^{0\leq}) \cong \widetilde{\operatorname{vTr}}(\mathcal{U}_Q(\mathfrak{gl}_m)^{0\leq}),$$

and, in particular, an equivalence of categories

$$\operatorname{hTr}(n\operatorname{Foam}) \cong \widetilde{\operatorname{vTr}}(n\operatorname{Foam}).$$

The preceding result establishes the analogous equivalence after passing to homotopy complexes. The conjecture strengthens this to an equivalence before forming complexes and reflects the observed fact that annular web closures evaluate as nonnegative Laurent-polynomial combinations of nested circles.