This entire question takes place in the $HF_p$-local category of $p$-local spectra, i.e. the essential image of $HF_p$-localization on the stable homotopy category. $HF_p$ itself is in there, and of course so is the $HF_p$-local sphere $L_{HF_p}S^0$. Let $loc(X)$ denote the smallest localizing subcategory containing an object $X$.

**Question**: Is there a way to show that $L_{HF_p}S^0$ is NOT in $loc(HF_p)$?

I'm imagining there might be some property $HF_p$ has that's preserved by triangles and coproducts, but that $L_{HF_p}S^0$ does not have. (But remember that coproducts here are not the same as in the full category of spectra; likewise the smash product).

Here's why I'm interested. If the answer is yes, then this local category has at least three localizing subcategories: $loc(0)$, $loc(HF_p)$, and $loc(L_{HF_p}S^0)$, the last one being the entire category. (I'm pretty sure $loc(HF_p)$ is minimal, but that's irrelevant.) This would be exciting, because I recently found out that the Bousfield lattice of this category has exactly two elements. If the answer to my question above is yes, then there's a localizing subcategory, namely $loc(HF_p)$, that isn't a Bousfield class. (The only other time that this is known to happen is in the derived category of a certain bizarre type of non-Noetherian ring; see Stevenson's http://www.arxiv.org/abs/1210.0399).

If one took a $K(n)$ instead of $HF_p$, the answer to my question is no. Hovey and Strickland ("Morava $K$-theories and localization") classified localizing subcategories in the $K(n)$-local category, and there are only two (ditto with Bousfield classes). But their proof uses that $L_{K(n)}F(n)$ is a small graded weak generator in the $K(n)$-local category. Later in the same paper they show that the $HF_p$-local category has no nonzero small objects, so the same trick won't work.

Or, on the other hand, if anyone knows how to build $L_{HF_p}S^0$ from $HF_p$, that's great too. But that seems harder.