# Lower bounds on “size” of Whitehead covers?

Let $$X$$ be a nonzero finite spectrum, connective say, and consider the Whitehead tower of $$n$$-connected covers $$\dots \to X\langle n \rangle \to X\langle n-1 \rangle \to \dots \to X\langle 0 \rangle = X$$.

Question: Do the spectra $$X\langle n \rangle$$ grown in "size" as $$n$$ increases?

By "size", I mean something like "number of cells" (in a minimal cell structure). As a variant, I'd be happy to understand how the Poincare series $$\sum_i \dim H_i(X\langle n \rangle) z^i$$ (with $$\mathbb F_p$$ coefficients, say) "grows", where now there are several notions of "growth" which might be relevant.

I've restricted to nonzero finite spectra $$X$$ to rule out bounded-above spectra, for which $$X\langle n \rangle$$ is eventually zero. My hunch is that with this restriction, $$X\langle n \rangle$$ at the very least can't decay in size too quickly.

I'd also be interested in the unstable case -- though we should probably assume that $$X$$ is simply-connected to rule out pathologies.

One can attain upper bounds on the grown in size of $$X \langle n \rangle$$ using the cofiber sequence $$X \langle n \rangle \to X\langle n-1 \rangle \to \Sigma^n H\pi_n(X)$$. I'm more interested in lower bounds, though.