I have often heard that it is not known whether or not the Brown-Peterson spectrum $BP$ is an $E_\infty$-ring spectrum. Though I see that this is a somewhat natural question to ask, I have often wondered what would the practical consequences of an answer to this question would be.
In particular, I am wondering what would a positive answer to this question mean for homotopy theory. Are there computations we would be able to do that we couldn't do before? What would be the take away if $BP$ did not admit an $E_\infty$-ring structure?
I would also appreciate any references on this problem. Thanks!