The conjectural abelian variety attached to the sign eigenspaces

Let $\ell$ be a prime, let $K_\ell$ be the unramified quadratic extension of $\mathbb{Q}_\ell$, and let $T_\epsilon/L_\epsilon$ be the rigid analytic quotient associated with the sign $\epsilon\in\{+,-\}$, where $T_\epsilon=\mathbb{G}_m\otimes_\mathbb{Z}H_\epsilon$. The sign-eigenspace isogeny conjecture. For each $\epsilon\in\{+,-\}$, there is an abelian variety $J^{(\ell)}_\epsilon$ defined over $\mathbb{Q}$ such that $T_\epsilon/L_\epsilon$ is isomorphic over $K_\ell$ to the rigid analytic space associated with $J^{(\ell)}_\epsilon$. This is stated as a variant of a conjecture of Bertolini, Darmon and Dasgupta and supplies the abelian varieties used later to formulate algebraicity properties of Darmon points; no resolution is given in the source.