Logarithmic-dimension conjecture for shifted algebraic numbers

Let $\alpha$ be an algebraic number of degree $d\geq2$. Consider the rational vector space spanned by the logarithms of its integer shifts, $\operatorname{span}_{\mathbb Q}\{\log(n+\alpha):0\leq n<x\}$. Logarithmic-dimension conjecture.

$$\dim \operatorname{span}_{\mathbb Q}\{\log(n+\alpha):0\leq n<x\}\sim(1-\rho(d))x.$$

This is presented as a plausible quantitative conjecture suggested by the same reasoning used earlier for multiplicative dependence. The source provides no resolution, so the asymptotic remains open.