The spectral Sullivan conjecture on symmetric-power layers

Let $p$ be a prime, let $\mathbb{S}_{(p)}$ denote the $p$-local sphere spectrum, and let $\mathrm{Sp}^{p^k}$ denote the $p^k$th symmetric power. Define

$$L(k)=\Sigma^{-k}\mathrm{Sp}^{p^k}/\mathrm{Sp}^{p^{k-1}}\mathbb{S}_{(p)}.$$

For a spectrum $X$, write $\langle X\rangle^s$ for its connective Bousfield class, the subcategory generated under colimits and extensions by $X$.

The spectral Sullivan conjecture. The connective Bousfield class of $L(k)$ is the same as that of some type $k$ finite spectrum.

This proposed generalization arises from the symmetric power filtration and concerns the chromatic complexity of the layers $L(k)$. The supplied text gives no resolution or further evidence for the claim, so its status remains open.