Colored Rasmussen spectral sequence conjecture for the unknot

Let $\mathrm{HHH}_{S^n}(O_1)$ be the $S^n$-colored triply graded Khovanov–Rozansky homology of the unknot, with even generators $u_0,\ldots,u_{n-1}$ and odd generators $\xi_0,\ldots,\xi_{n-1}$. Let $\mathrm{Kh}_{S^n}(O_1)$ denote the $S^n$-colored Khovanov homology. Colored Rasmussen spectral sequence conjecture. The Rasmussen spectral sequence from $\mathrm{HHH}_{S^n}(O_1)$ to $\mathrm{Kh}_{S^n}(O_1)$ has only one nontrivial differential $d_2$, given by

$$d_2(u_i)=0,\qquad d_2(\xi_i)=\sum_{j=0}^{i}u_ju_{i-j}.$$

Consequently,

$$\mathrm{Kh}_{S^n}(O_1)\simeq H^*\left(\mathbb{Z}[u_0,\ldots,u_{n-1},\xi_0,\ldots,\xi_{n-1}],d_2\right).$$

This is proposed as the colored extension of Rasmussen's spectral sequence, identifying colored Khovanov homology of the unknot with the cohomology of the stated differential graded algebra.