Schinzel's bounded orthogonal relation conjecture

Let $F,G\in \mathbb{Z}[x_1,\dots,x_n]$ be relatively prime. Let ${\bf a}:=(a_1,\dots,a_n)\in\mathbb{Z}^n$ and let $\xi\in\overline{\mathbb{Q}}^*$ be not a root of unity. Suppose that

$$F(\xi^{a_1},\dots,\xi^{a_n})=G(\xi^{a_1},\dots,\xi^{a_n})=0.$$

Schinzel's conjecture. There exists a nonzero vector ${\bf b}\in\mathbb{Z}^n$ orthogonal to ${\bf a}$ such that

$$\lVert{\bf b}\rVert_{\infty}\leq B(F,G),$$

where $B(F,G)>0$ depends only on $F$ and $G$. The conjecture predicts a uniformly bounded multiplicative relation among the exponents whenever the two relatively prime polynomials vanish at the corresponding powers of a non-torsion algebraic number. The paper's abstract states that it gives a new proof of this conjecture and an explicit result depending on the height and degree of the variety, so the parser's unknown status should be checked against the paper's resolution.