The Khovanov-basicity conjecture for alternating links

Let $L$ be a link with $c$ components. Call $L$ Khovanov-basic if its Khovanov complex $\operatorname{Kh}(L)$ is homotopy equivalent to a direct sum of basic complexes and exactly $2^{c-1}$ one-term basic complexes $\Omega_1$ occur, where, up to degree and height shifts, $\Omega_1$ is the one-term complex whose only nonzero term is a smoothing consisting of a single circle, and $\Omega_2$ is the two-term complex whose two nonzero terms are such smoothings and whose differential is the genus-one surface with one circle boundary at the top and one at the bottom. Khovanov-basicity conjecture. Every alternating link is Khovanov-basic. This would provide a structural explanation for the observed Betti-number patterns of alternating links, although the text notes that some links are not Khovanov-basic and does not claim that all links have this property.