The Brown-Peterson spectrum is not $E_{2(p^2+2)}$ at odd primes

Recently, Lawson has shown that the 2-primary Brown-Peterson spectrum does not admit the structure of an $E_{12}$ ring spectrum, thus answering a question of May in the negative. We extend Lawsons result to odd primes by proving that the p-primary Brown-Peterson spectrum does not admit the structure of an $E_{2(p^2+2)}$ ring spectrum. We also show that there can be no map $MU to BP$ of $E_{2p+3}$ ring spectra at any prime.