Amoroso–David geometric essential minimum conjecture for torus translates

Let $X\subset \mathbf{G}_m^N(\overline{\mathbf{Q}})$ be a variety, and let $V_X$ be the smallest translate of a subtorus containing $X$. Write $\widehat{\mu}^{\mathrm{ess}}(X)$ for the essential minimum of the normalized Weil height, $\deg(V_X)$ for the degree of $V_X$, and $\omega_{\overline{\mathbf{Q}}}(X;V_X)$ for the obstruction degree of $X$ in $V_X$. Amoroso–David's geometric conjecture. There exists a real constant $c(N)>0$ such that

$$\widehat{\mu}^{\mathrm{ess}}(X)\geq c(N)\frac{\deg(V_X)}{\omega_{\overline{\mathbf{Q}}}(X;V_X)}.$$

The source presents this as the geometric analogue of the preceding conjecture, expected to be especially relevant for translates of subtori by points of infinite order; it gives no resolution status.