Surjectivity conjecture for higher pages of cyclic equivariant slice spectral sequences

For $m,n\geq 1$, let

$$\operatorname{SliceSS}(BP^{(C_{2^n})}\langle m+1\rangle)\longrightarrow \operatorname{SliceSS}(BP^{(C_{2^n})}\langle m\rangle)$$

be the map of slice spectral sequences induced by the quotient map. Write $\mathcal{E}_r$ for the $r$-th page and $d_r$ for its differential.

Surjectivity conjecture. For all $m\geq 1$ and $r\geq 2$, this map induces a surjection on the $\mathcal{E}_r$-page and on the $d_r$-differentials.

Surjectivity on the $\mathcal{E}_2$-page is known and is suggested as a route toward proving the height classification conjecture. Surjectivity on every higher page and for the corresponding differentials is not established in the supplied context.