Isotopy invariance of collapsed generalized reduced Khovanov homology in $S^2 \times S^1$

Let $T$ be an oriented 1-tangle diagram in the annulus, and let $L_{T^0}$ be the link in $S^2 \times S^1$ obtained by closing it with an overpass or underpass arc. Let $(C_0,\partial_0)$ be the associated cochain complex, and denote by $\operatorname{Khr}(S^2 \times S^1,T)$ its cohomology with the bigradings collapsed from $\mathbbm{Z}$ to $\mathbbm{Z}_2$. The isotopy-invariance conjecture. If $L_{T_1^0}$ and $L_{T_2^0}$ are isotopic links in $S^2 \times S^1$, then

$$\operatorname{Khr}(S^2 \times S^1,T_1)=\operatorname{Khr}(S^2 \times S^1,T_2).$$

The claim would make the generalized reduced Khovanov homology an invariant of links in $S^2 \times S^1$, rather than merely of their tangle-diagram descriptions. The authors report that ordinary bigraded homology can depend on the description, while the checked examples agree after collapsing to $\mathbbm{Z}_2$; the conjecture remains open.