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We show that the odd-primary Brown-Peterson spectrum $\mathrm{BP}$ does notadmit the structure of an $\mathbb{E}_{2(p^2+2)}$ ring spectrum and that therecan be no map $\mathrm{MU} \to \mathrm{BP}$ of $\mathbb{E}_{2p+3}$ ringspectra. We also prove the same results for truncated Brown-Peterson spectra$\mathrm{BP} \langle n \rangle$ of height $n \geq 4$. This extends results ofLawson at the prime $2$.

The Brown-Peterson spectrum is not $\mathbb{E}_{2(p^2+2)}$ at odd primes