Finite-rank Lehmer conjecture for products of tori and abelian varieties

Let $A$ be an abelian variety defined over a number field $k$, and set $\mathbb{G}=\mathbb{G}_m^n\times A$. For a finite-rank subgroup $\Gamma\subset\mathbb{G}(\overline{k})$, let $\Gamma_\mathrm{sat}$ denote its saturation under endomorphisms and multiplication, and let $k(\Gamma)$ be its field of rationality. For $(\boldsymbol{\alpha},P)\in\mathbb{G}(\overline{k})$, define

$$\hat{h}_\mathbb{G}(\boldsymbol{\alpha},P):=\hat{h}_{\mathbb{G}_m^n}(\boldsymbol{\alpha})+\hat{h}_A(P).$$

Finite-rank Lehmer conjecture for products. There exists $c_\Gamma>0$ such that

$$\hat{h}_\mathbb{G}(\boldsymbol{\alpha},P)\geq c_\Gamma$$

for every $(\boldsymbol{\alpha},P)\in\mathbb{G}(k(\Gamma))\backslash\Gamma_\mathrm{sat}$. The preceding conjecture fails in general for semi-abelian varieties because of non-torsion Ribet points, but the paper explains why products of a torus and an abelian variety avoid this obstruction. The conjecture is presented as a broader form of the finite-rank statement above and is open.