The differential-generation conjecture for the homological slice spectral sequence of $BPGL\langle m\rangle$

Let $BPGL\langle m\rangle$ be the truncated motivic Brown–Peterson spectrum, and let its homological slice spectral sequence (HSSS) have differentials $d_r$. The differentials displayed in Theorem are the differentials

$$d_{2^{i+1}-1}(\zeta_j^{2^{i+1-j}})=\overline{v}_{i}\rho^{2^i-1}\left(p_{j-1}\left(\frac{x_1}{\rho},\ldots,\frac{x_{j-1}}{\rho}\right)\right)^{2^{i+1-j}}$$

for $1\leq i\leq m$ and $1\leq j\leq i+1$, where $p_j$ is determined by $\zeta_j=p_j(\xi_1,\ldots,\xi_j)$ through the inversion formulas in the Hopf algebra $\mathcal A_*$. Differential-generation conjecture. All differentials in the HSSS for $BPGL\langle m\rangle$ are generated under the Leibniz rule by those in Theorem. In particular, the spectral sequence collapses on $E_{2^{m+1}}$. This conjecture would make the explicitly known family of differentials sufficient to determine the entire differential structure of the spectral sequence, whose general behavior is otherwise understood only through the stated differential and edge theorems.