Uniqueness of $E_\infty$ structures for connective covers

We refine our earlier work on the existence and uniqueness of $E_\infty$ structures on $K$-theoretic spectra to show that the connective versions of real and complex $K$-theory as well as the connective Adams summand $\ell$ at each prime $p$ have unique structures as commutative $\mathbb{S}$-algebras. For the $p$-completion $\ell_p$ we show that the McClure-Staffeldt model for $\ell_p$ is equivalent as an $E_\infty$ ring spectrum to the connective cover of the periodic Adams summand $L_p$. We establish a Bousfield equivalence between the connective cover of the Lubin-Tate spectrum $E_n$ and $\mathit{BP}\langle n\rangle$.