Khovanov's alternating-knot refinement

Let $L$ be a prime alternating knot, and let $s(L)$ and ${\text{\it Kh}}'(L)$ be the integer and Laurent polynomial from Khovanov's structural conjecture. The signature of $L$ is the usual knot signature, and a polynomial depends only on $tq^2$ when it is a polynomial in that single monomial. Khovanov's alternating-knot refinement. The integer $s(L)$ is equal to the signature of $L$, and ${\text{\it Kh}}'(L)$ contains only powers of $tq^2$.

This refinement predicts that the structural decomposition is controlled by the signature and that the remaining factor is supported on a single diagonal for prime alternating knots. It is stated as a computationally motivated conjecture in the paper, with no resolution supplied there.