Amoroso–David's obstruction-index conjecture for multiplicatively independent points

Let $n$ be a positive integer and let $\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_n)\in\mathbb{G}_m^n(\overline{\mathbb{Q}})$ have multiplicatively independent coordinates. Let $\delta(\boldsymbol{\alpha})$ be the degree of a nonzero polynomial in $\mathbb{Q}[x_1,\ldots,x_n]$ of minimal degree vanishing at $\boldsymbol{\alpha}$, and let $h(\boldsymbol{\alpha})$ denote its height.

Amoroso–David's obstruction-index conjecture. There exists a real number $c(n)>0$ such that

$$h(\boldsymbol{\alpha})\geq\frac{c(n)}{\delta(\boldsymbol{\alpha})}.$$

This conjecture is the point-height form of the multivariable Lehmer problem. The source presents it as a conjecture of Amoroso and David; its status is open.