Fix a prime $p$. Let $M_p(i)$, the $i$-th Moore spectrum at the prime $p$, be the cofiber of the map $$ S^0 \overset{p^i}\longrightarrow S^0 $$ where $S^0$ be the sphere spectrum. In the Mathoverflow discussion, it was pointed out that $M_p(i)$ is not $E_{\infty}$. Reference to a proof was given.

Is this the first/only known proof of Moore spectrum $M_p(i)$ (where $i> 1$ ) not being $E_{\infty}$? If not, when was it first proved and by whom (Please include references if possible)? Feel free to write /refer to a proof which is different from the one that is given here.