Manin–Mumford conjecture for abelian uniformizable $T$-modules

Let $A$ be the coefficient ring and let $B$ be a subring of $A$. Let $\mathcal{A}=(\mathbb{G}_{a}^{m},\Phi)$ be an abelian uniformizable $T$-module, meaning an abelian uniformizable $T$-module as in the paper, and suppose that there exists $i\in\mathbb{N}-\{0\}$ such that the leading coefficient of $\Phi(T^{i})$ is an invertible matrix. Let $X$ be an algebraic subvariety of $\mathcal{A}$ that contains no translate by a torsion point of $\mathcal{A}$ of any nontrivial sub-$B$-module of $\mathcal{A}$, for any subring $B$ of $A$. Manin–Mumford conjecture. The variety $X$ contains only finitely many torsion points of $\mathcal{A}$. This proposes a generalization of the Manin–Mumford principle to abelian and uniformizable $T$-modules. Earlier naive formulations are known to fail, motivating the inclusion of all sub-$B$-modules and the stronger finiteness hypotheses in this statement.