Paulin's geometric level raising conjecture for the eigencurve

Fix distinct primes $p$ and $l$, and an integer $N$ coprime to $pl$. Let $\mathcal{E}$ be the cuspidal eigencurve of tame level $\Gamma_0(Nl)$, parametrising overconvergent cuspidal $p$-adic modular eigenforms. For a point $\phi$ of $\mathcal{E}$ corresponding to an eigenform $f_\phi$, let $\pi_{f_\phi,l}$ be the associated representation of $\mathrm{GL}_2(\mathbb{Q}_l)$. An irreducible connected component $\mathcal{Z}$ of $\mathcal{E}$ is generically unramified principal series if the representations associated to its points away from a discrete set are unramified principal series, and generically special if they are special away from a discrete set. Let $\alpha$ and $\beta$ be the roots of

$$X^2-t_lX+ls_l,$$

where $t_l$ and $s_l$ are the $T_l$ and $S_l$ eigenvalues of $f_\phi$. Paulin's geometric level raising conjecture. If $\mathcal{Z}$ is generically unramified principal series and there is a point $\phi$ on $\mathcal{Z}$ where the ratio of $\alpha$ and $\beta$ is $l^{\pm 1}$ and $\pi_{f_\phi,l}$ is special, then there exists a generically special component $\mathcal{Z'}$ intersecting $\mathcal{Z}$ at $\phi$. This predicts that a special representation at a level-raising point lies on a component whose generic local representation is special, extending geometric level raising phenomena on the eigencurve beyond the cases established by the paper's theorem.