Redshift and multiplication for truncated Brown-Peterson spectra

We equip $\mathrm{B}\mathrm{P}⟨n⟩$ with an ${\mathbb{E}}_3$-$\mathrm{B}\mathrm{P}$-algebra structure, for each prime $p$ and height $n$. The algebraic $K$-theory of this ring is of chromatic height exactly $n+1$, and the map $$\mathrm{K}$($\mathrm{B}\mathrm{P}$\langle n \rangle){(p)} \to $\mathrm{L}${n+1}^{f} $\mathrm{K}$($\mathrm{B}\mathrm{P}$\langle n\rangle)_{(p)}$ has bounded above fiber.