The Knight Move Conjecture for Khovanov homology

Let $K$ be a knot. Its Khovanov homology over $\mathbb{Q}$ is a bigraded vector space, and write $s$ for Rasmussen's invariant. A pawn move piece is

$$\mathbb{Q}\{0,s-1\}\oplus\mathbb{Q}\{0,s+1\},$$

and a knight move piece is

$$\mathbb{Q}\{i,j\}\oplus\mathbb{Q}\{i+1,j+4\},$$

for $i,j\in\mathbb{Z}$. Knight Move Conjecture. For every knot $K$, its Khovanov homology over $\mathbb{Q}$ is the direct sum of one pawn move piece and several knight move pieces. This conjecture holds for knots whose Lee spectral sequence degenerates after the first page, including alternating, quasi-alternating, and knots with unknotting number at most $2$, but the paper presents a counterexample: in general the Lee spectral sequence can have a nontrivial differential of bidegree $(1,8)$.