The two-point thin-filling conjecture for tangles

Let $T$ be a tangle. Write $\operatorname{ThinHF}(T)$ and $\operatorname{ThinKh}(T)$ for the spaces of thin fillings associated respectively to Heegaard Floer homology and Khovanov homology. The two-point thin-filling conjecture. There is no tangle $T$ such that

$$\operatorname{ThinHF}(T)\text{ or }\operatorname{ThinKh}(T)$$

consists of two points. The paper notes that all other types of thin rational filling spaces described earlier occur in examples, while it does not know whether the additional case of precisely two distinct thin rational fillings occurs.