Distinguished-component independence conjecture for Szabó's reduced spectral sequence

Let $L$ be a link, and let $c$ and $c'$ be components of $L$. Denote by $\widetilde{E}^k(L,c)$ the reduced Szabó spectral sequence associated to $L$ with distinguished component $c$. Distinguished-component independence conjecture. For $k\geq 2$,

$$\widetilde{E}^k(L,c)\cong\widetilde{E}^k(L,c').$$

If true, the reduced spectral sequence could be denoted simply by $\widetilde{E}^k(L)$, independently of the distinguished component. The conjecture was verified in the computations reported for the specified knot and link tables.