  We equip $\mathrm{BP} \langle n \rangle$ with an
$\mathbb{E}_3$-$\mathrm{BP}$-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 $\mathrm{K}(\mathrm{BP}\langle n \rangle)_{(p)} \to
\mathrm{L}_{n+1}^{f} \mathrm{K}(\mathrm{BP}\langle n\rangle)_{(p)}$ has bounded
above fiber.
