Rank-three Khovanov homology classification for null-homologous knots in $\mathbb{RP}^3$

Let $K$ be a null-homologous knot in $\mathbb{RP}^3$. Write $\widetilde{\operatorname{Kh}}(K;\mathbb{Z}/2)$ for its reduced Khovanov homology over $\mathbb{Z}/2$. Rank-three Khovanov homology conjecture. If

$$\dim\widetilde{\operatorname{Kh}}(K;\mathbb{Z}/2)=3,$$

then $K$ is either the local trefoil or the knot $2_1$ in the knot table. This would classify the null-homologous knots in $\mathbb{RP}^3$ with reduced Khovanov homology of rank $3$; the paper gives no resolution of the conjecture.