  We show that the odd-primary Brown-Peterson spectrum $\mathrm{BP}$ does not
admit the structure of an $\mathbb{E}_{2(p^2+2)}$ ring spectrum and that there
can be no map $\mathrm{MU} \to \mathrm{BP}$ of $\mathbb{E}_{2p+3}$ ring
spectra. We also prove the same results for truncated Brown-Peterson spectra
$\mathrm{BP} \langle n \rangle$ of height $n \geq 4$. This extends results of
Lawson at the prime $2$.
