The exceptional-set power conjecture for arithmetic zeta functions

Let $\gamma$ be a complex number and let $S_{\gamma}$ denote the set of natural numbers $n$ such that $n^{\gamma}$ is algebraic. If $\gamma\notin\mathbb{Q}$, then there exists $B\in\mathbb{N}$ such that

$$S_{\gamma}=\{B^k\mid k\geq 0\}.$$

Exceptional-set power conjecture. If $\gamma\notin\mathbb{Q}$, then there exists $B\in\mathbb{N}$ such that

$$S_{\gamma}=\{B^k\mid k\geq 0\}.$$

This conjecture predicts that, outside the rational case, the integers whose $\gamma$-th powers are algebraic form exactly the multiplicative semigroup generated by one integer. It would in particular determine the example $S_{\log 3/\log 2}$ as $\{1,2,4,8,\ldots\}$; the paper's preceding proposition only establishes that any three elements of $S_{\gamma}$ are multiplicatively dependent when $\gamma$ is transcendental.