The Brown-Peterson spectrum is not E2(p2+2) at odd primes

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