Amoroso–David essential minimum conjecture over the rationals

Let $X\subset \mathbf{G}_m^N(\overline{\mathbf{Q}})$ be a variety, and let $U_X^{\mathbf{Q}}$ be the smallest union of torsion varieties containing $X$ that is defined over $\mathbf{Q}$. Here $\widehat{\mu}^{\mathrm{ess}}(X)$ denotes the essential minimum of the normalized Weil height, $\deg(U_X^{\mathbf{Q}})$ its degree, and $\omega_{\mathbf{Q}}(X;U_X^{\mathbf{Q}})$ the obstruction degree of $X$ in $U_X^{\mathbf{Q}}$, with the convention that the quotient is $0$ when $\dim(X)=\dim(U_X^{\mathbf{Q}})$. Amoroso–David's conjecture. There exists a real constant $c(N)>0$ such that

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

This is a conjectural lower bound for the height of Zariski-dense algebraic points, motivated by Lehmer's problem and generalized Bogomolov problems. Its formulation is attributed to Amoroso and David; the supplied text gives no resolution status.