The rank invariance conjecture for strong inversions

Let $K$ be a knot, and let $h_1$ and $h_2$ be strong inversions on $K$. Rank invariance conjecture. The dimensions of the associated equivariant Khovanov homology groups are equal:

$$\dim\varkappa(K,h_1)=\dim\varkappa(K,h_2).$$

This is based on observations from examples and would explain equal ranks arising from distinct strong inversions and corresponding branch sets; it remains open.