Localizing the $E_2$ page of the Adams spectral sequence

There is only one nontrivial localization of $\pi_*S_{(p)}$ (the chromatic localization at $v_0=p$), but there are infinitely many nontrivial localizations of the Adams $E_2$ page for the sphere. The first non-nilpotent element in the $E_2$ page after $v_0$ is $b_{10}\in \mathrm{Ext}_A^{2p(p-1)-2}(\mathbb{F}_p,\mathbb{F}_p)$. We work at $p=3$ and study $b_{10}^{-1}\mathrm{Ext}_P(\mathbb{F}_3,\mathbb{F}_3)$ (where $P$ is the algebra of dual reduced powers), which agrees with the infinite summand $\mathrm{Ext}_P(\mathbb{F}_3,\mathbb{F}_3)$ of $\mathrm{Ext}_A(\mathbb{F}_3,\mathbb{F}_3)$ above a line of slope ${1\over 23}$. We compute up to the $E_9$ page of an Adams spectral sequence in the category $\mathrm{Stable}(P)$ converging to $b_{10}^{-1}\mathrm{Ext}_P(\mathbb{F}_3,\mathbb{F}_3)$, and conjecture that the spectral sequence collapses at $E_9$. We also give a complete calculation of $b_{10}^{-1}\mathrm{Ext}_P^*(\mathbb{F}_3,\mathbb{F}_3[\xi_1^3])$.