Let $X$ be a finite spectrum, and let $N = dim_{\mathbb F_p} H_\ast(X;\mathbb F_p)$. I believe that $p$-completion $X^\wedge_p$ may be built as an $N$-cell complex where the cells are shifts of the $p$-complete sphere $S^\wedge_p$. That is, there is an $N$-step filtration $0 = X_0 \to X_1 \to \cdots \to X_N$ where the associated graded has a shift of $S^\wedge_p$ at each step.

Question 1: Let $X$ be a finite spectrum, and let $N = dim_{K(n)_\ast}K(n)_\ast X$. Then can the $K(n)$-localization $L_{K(n)} X$ be built as an $N$-cell complex where the cells are shifts of the $K(n)$-local sphere $L_{K(n)} S$?

Question 2: Same, but don't assume that $X$ is finite; only assume that $N$ is finite.

The spectra considered in Question 2 are the closure of those considered in Question 1 under retracts.

If it makes a difference to work $T(n)$-locally rather than $K(n)$-locally, that would be interesting.