Strict rank inequality between Khovanov and instanton Floer homology for non-local knots in $\mathbb{RP}^3$

Let $K$ be a non-local null-homologous knot in $\mathbb{RP}^3$. Let $\widetilde{\operatorname{Kh}}(K;\mathbb{Z}/2)$ denote reduced Khovanov homology over $\mathbb{Z}/2$, and let $\operatorname{I^{\natural}}(\mathbb{RP}^3,K;\mathbb{C})$ denote the corresponding instanton Floer homology. Khovanov–instanton rank inequality conjecture. Then

$$2\dim\widetilde{\operatorname{Kh}}(K;\mathbb{Z}/2)>\dim\operatorname{I^{\natural}}(\mathbb{RP}^3,K;\mathbb{C}).$$

The inequality expresses the heuristic that non-local knots have larger Khovanov-homology rank than instanton-homology rank. The paper presents it as an expected statement and gives no proof or resolution.