Vanishing of the tail-homology direct limit for non-B-adequate diagrams

Let $D$ be a link diagram. For each $N$, let $\widetilde{\mathrm{H}}^{\mathrm{Kh}}(\xDclv{N})$ be its shifted Khovanov homology, and let

$$\mathrm{H}^{\infty}(D)=\lim\limits_{\rightarrow}\widetilde{\mathrm{H}}^{\mathrm{Kh}}(\xDclv{N})$$

be the direct limit of the degree-preserving maps constructed in the paper. A diagram is B-adequate if it has the relevant B-adequate state property. Vanishing conjecture. If $D$ is not B-adequate, then

$$\mathrm{H}^{\infty}(D)=0.$$

This is the proposed complement to the stabilization theorem for B-adequate diagrams: the direct limit is finite-dimensional in every bi-degree, while the paper conjectures that it vanishes outside the B-adequate case. The source gives no resolution.