Equality conjecture for sumsets involving algebraic multipliers

Let $\alpha$ be a real or complex algebraic number. For an irreducible polynomial $f(x)\in\mathbb{Z}[x]$ with factorization

$$f(x)=\prod_{i=1}^d(a_i x+b_i),$$

define

$$H(f)=\prod_{i=1}^d(|a_i|+|b_i|).$$

For real $\alpha$ with minimal polynomial $f$, consider

$$\lim_{n\rightarrow\infty}\min_{A\subset\mathbb{R},\ |A|=n}\frac{|A+\alpha A|}{|A|}\leqslant H(f).$$

Equality conjecture. For any real algebraic $\alpha$, the inequality becomes an equality. For complex algebraic $\alpha$, the analogous equality holds for $A\subset\mathbb{C}$. This conjecture predicts that the upper bound obtained from the minimal polynomial exactly determines the asymptotic minimum growth of $A+\alpha A$; its status is not resolved in the supplied source.