The diagram-count bounds conjecture for diagramless link homology

Let $L$ be a non-split link and let $k$ be a fixed positive integer. Let $n_k(L)$ be the number of distinct $k$-crossing link diagrams for $L$ up to 2-space isotopy. Let $N$ be the number of copies of Khovanov homology in the diagramless homology of $L$ with $k$ crosscuts. Diagram-count bounds conjecture. One has

$$\frac{n_k(L)}{2}\leq N\leq n_k(L).$$

This conjecture relates the multiplicity of Khovanov homology in diagramless link homology to the number of link diagrams; the factor of two reflects the additional identification obtained by flipping diagrams. The source gives no resolution or further evidence.