The full Denis–Mordell–Lang conjecture for Drinfeld modules

Let $g$ be a positive integer, and let $\phi_1:A\to K\{\tau\},\dots,\phi_g:A\to K\{\tau\}$ be Drinfeld modules. An algebraic $(\phi_1,\dots,\phi_g)$-submodule of $\mathbb{G}_a^g$ is an irreducible algebraic subgroup invariant under the action of $(\phi_1,\dots,\phi_g)$. A $\phi$-module $M$ is finite rank if it contains a finitely generated $\phi$-submodule $M'$ such that $M/M'$ is a torsion $\phi$-module. Let $V\subset\mathbb{G}_a^g$ be an affine variety defined over $\overline{K}$, and let $\Gamma$ be a finite rank $(\phi_1,\dots,\phi_g)$-submodule of $\mathbb{G}_a^g(\overline{K})$. The full Denis–Mordell–Lang conjecture. There exist algebraic $(\phi_1,\dots,\phi_g)$-submodules $B_1,\dots,B_l$ of $\mathbb{G}_a^g$ and points $\gamma_1,\dots,\gamma_l\in\Gamma$ such that

$$V(\overline{K})\cap\Gamma=\bigcup_{i=1}^l(\gamma_i+B_i(\overline{K}))\cap\Gamma.$$

Denis showed that, under certain natural Galois-theoretic assumptions, this full conjecture would follow from a weaker conjecture concerning finitely generated $\phi$-modules. The case in which $\Gamma$ is the product of the torsion submodules of the $\phi_i$ was proved by Scanlon, and other instances are known, but the full statement remains unresolved.