As discussed for example in this paper (and this MO thread), there are obstructions for the existence of $A_n$-structures on Moore spectra $M(p^r):=\mathbb{S}/p^r$, which in general don't vanish. In particular, for odd primes, $M(p)$ does not admit an $A_p$-strucutre, and if I understand correctly, it is expected that for no value of $r$ the spectrum $M(p^r)$ admits an $A_\infty$-structure.

I am interested in the analogous question for the $d$-truncated Moore spectra $M(p^r,d) := \tau_{\le d}(\mathbb{S}/p^r)$:

What is known about the existence of $A_n$ (or perhaps even $E_n$) structures on $M(p^r,d)$?

For $d=0$ one gets $M(p^r,0) \simeq H\mathbb{Z}/p^r$, which is an $E_\infty$-ring. However, I am not sure what is the situation even for $d=1$. Specifically, I am interested in the case where $p$ and $d$ are fixed and $r \gg 0$. It would be best if one could get an $E_\infty$-strucutre (the goal being to approximate $\tau_{\le d}(\mathbb{S}_p)$ by $\pi$-finite $E_\infty$-rings).