The characteristic-zero comparison conjecture for symplectic annular Khovanov homology

Let $L$ be an annular link, let $k\in\mathbb Z$ be its winding grading, and let $\mathbb F$ be a field of characteristic zero. Denote by $\operatorname{AKh}_{symp}^i(L;k;\mathbb F)$ the symplectic annular Khovanov homology in absolute homological grading $i$, and by $\operatorname{AKh}^{j,q}(L;k;\mathbb F)$ the combinatorial annular Khovanov homology in homological grading $j$, quantum grading $q$, and winding grading $k$. The characteristic-zero comparison conjecture. There is an isomorphism

$$\operatorname{AKh}_{symp}^i(L;k;\mathbb F)\simeq\bigoplus_{j-q=i}\operatorname{AKh}^{j,q}(L;k;\mathbb F).$$

If $\operatorname{AKh}_{symp}(L)$ is equipped with a relative quantum equivariant grading from a non-commutative vector field, this isomorphism may be upgraded to an isomorphism of relatively trigraded groups. The claim is presented as an expected relationship, analogous to the non-annular comparison of Abouzaid and Smith, and the stronger trigraded refinement depends on the additional equivariant grading.