Height classification conjecture for differentials in cyclic equivariant slice spectral sequences

For $n,m\geq 1$, consider the $C_{2^n}$-equivariant spectrum $BP^{(C_{2^n})}$ and its truncation $BP^{(C_{2^n})}\langle m\rangle$, together with their associated $C_{2^n}$-slice spectral sequences. Suppose the differentials in the slice spectral sequence of $BP^{(C_{2^n})}$ are classified by height.

Height classification conjecture. The $C_{2^n}$-slice spectral sequence of $BP^{(C_{2^n})}\langle m\rangle$ contains all differentials in the slice spectral sequence of $BP^{(C_{2^n})}$ having height at most

$$h=2^{n-1}m.$$

This conjecture would give a transchromatic method for comparing fixed-point computations across heights. The paper notes that the map from the truncation at $m+1$ to the truncation at $m$ is surjective on the $[... E_2$-page, but the asserted control of all differentials remains open.