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

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