If $X$ is a spectrum, we have a notion of its connective part $X_{\le 0}$ and the corresponding notion of truncation $X_{[i:j]} = X_{\le j}/X_{\le i-1}$, where $X_{\le j}$ is deduced from $X_{\le 0}$ using the shift functor and the quotient above is defined as a cone. * Note that I'm using cohomological grading. *

I know that the functor $X_{\le 0}$ is symmetric monoidal (with respect to the smash product), and in particular takes $E_n$ ring spectra to $E_n$ ring spectra. On the other hand, the functor $X_{[0:0]}$ also takes $E_n$ ring spectra to $E_n$ ring spectra (in fact, even to $E_\infty$ ring spectra if $n\ge 2$) since $X_{[0:0]}$ is the Eilenberg-Maclane spectrum associated to $\pi_0(X)$.

My question is: when is it true that the truncation functor $X_{[i:j]}$ takes $E_n$ ring spectra to $E_n$ ring spectra? What if we assume $X$ is (co)connective?