Let $M_p(i)$ be the mod $p^i$ Moore spectrum, i.e. the cofiber of $p^i: \mathbb S \to \mathbb S$. Upper and lower bounds on the $n$ for which $M_p(i)$ admits an $A_n$ structure are known, cf. Bhattacharya. I gather from this that $M_p(i)$ admits at least an $A_2$ structure for all primes $p$ and $i \in \mathbb N$, except for the mod-2 Moore spectrum $M_2(1)$, which does not admit an $A_2$ structure.

One consequence of a spectrum $X$ having an $A_2$ structure is that $X$ is a retract of $X\wedge X$. If $M_2(1)$ were a retract of $M_2(1) \wedge M_2(1)$, then the retract map would be an $A_2$ structure, so that can't happen.

But the Spanier-Whitehead dual of $M_p(i)$ is $\Sigma^{-1} M_p(i)$, so by a triangle equation we have that $M_p(i)$ is always a retract of $\Sigma^{-1} M_p(i)^{\wedge 3}$.

So it seems like there is conflicting evidence for the resolution of the following

Question: Is the mod-2 Moore spectrum $M_2(1)$ a retract of $\Sigma^n M_2(1) \wedge M_2(1)$ for some $n \in \mathbb Z$?