Galois-orbit equidistribution conjecture for torsion points and Laurent polynomials

Let $d\geq 1$, let $P\in \overline{\mathbb{Q}}[X_1^{\pm 1},\ldots,X_d^{\pm 1}]\setminus\{0\}$, and let $m(P)$ denote the logarithmic Mahler measure of $P$. For $\boldsymbol{\zeta}\in\mathbb{G}_m^d$ of finite order, let $\delta(\boldsymbol{\zeta})$ denote the minimum degree of a coordinate of $\boldsymbol{\zeta}$ over a proper algebraic subtorus, as defined in the paper. Galois-orbit equidistribution conjecture. For each $P$ there exists $\kappa>0$ such that, whenever $\delta(\boldsymbol{\zeta})$ is sufficiently large, one has $P(\boldsymbol{\zeta}^\sigma)\neq 0$ for every $\sigma\in\operatorname{Gal}(\mathbb{Q}(\boldsymbol{\zeta})/\mathbb{Q})$ and

$$\frac{1}{[\mathbb{Q}(\boldsymbol{\zeta}):\mathbb{Q}]}\sum_{\sigma\in\operatorname{Gal}(\mathbb{Q}(\boldsymbol{\zeta})/\mathbb{Q})}\log|P(\boldsymbol{\zeta}^\sigma)|=m(P)+O(\delta(\boldsymbol{\zeta})^{-\kappa})$$

as $\delta(\boldsymbol{\zeta})\to\infty$, with the implicit constant depending only on $d$ and $P$. The conjecture extends the quantitative torsion-point estimate of Lind, Schmidt, and Verbitskiy from finite subgroups to individual Galois orbits. The case $d=1$ follows from work of Baker, Ih, and Rumely, while the stated two-dimensional example remains open.