Uniform upper bound for accumulation exponents outside exceptional algebraic pairs

Let $\alpha$ be a nonzero real algebraic number and let $\zeta$ be real. Exclude pairs of the form

$$(\alpha,\zeta)=\left(\frac{A}{B}M^{-\frac{g}{L}},M^{\frac{1}{L}}\right),$$

where $L,M,A,B,g>0$ are integers. The quantity $\overline{\sigma}(\alpha,\zeta)$ is the upper rate of accumulation of $\alpha\zeta^n$ modulo $1$ to $0$. Uniform-bound conjecture. For every such pair,

$$\overline{\sigma}(\alpha,\zeta)\leq 1.$$

In particular, for any real algebraic $\zeta$ not of the form $\sqrt[L]{M}$ with $L,M$ integers, and any nonzero real algebraic $\alpha$, one has $\overline{\sigma}(\alpha,\zeta)\leq 1$. The preceding theorem proves only finiteness of the upper exponent under the same exceptional-form exclusion; the conjecture proposes the uniform bound $1$.