The structural decomposition conjecture for equivariant Khovanov homology

Let $(K,h)$ be a strongly invertible knot. For some $\delta\in\mathbb{Z}$, let

$$V=\mathbb{F}^{(0,\delta)}\oplus\mathbb{F}^{(2,\delta)}\oplus\mathbb{F}^{(3,\delta)}\oplus\mathbb{F}^{(5,\delta)}.$$

Here the second grading is regarded as a relative $\mathbb{Z}$-grading, and $V[m,n]$ denotes the corresponding shift by $(m,n)$. Structural decomposition conjecture. For every strongly invertible knot $(K,h)$, there are pairs $(m_i,n_i)\in\mathbb{Z}\times\mathbb{Z}$ such that

$$\varkappa(K,h)\cong\bigoplus_{i=1}^k V[m_i,n_i]$$

as a $(\mathbb{Z}\times\mathbb{Z})$-graded group, with the secondary grading relative. In particular,

$$\dim\varkappa(K,h)\equiv 0\bmod{4}.$$

The conjecture is motivated by empirical calculations and would explain the observed divisibility and ordered grading behaviour; no theoretical explanation or proof is given, so the claim remains open.