Krachun--Petrov conjecture on sums with an algebraic multiplier

Let $\lambda$ be a real algebraic number. Write its minimal polynomial as $f(x)=\prod_{i=1}^d(a_i x+b_i)$ over the complex numbers, with coprime integer coefficients, and define $H(\lambda)=\prod_{i=1}^d(|a_i|+|b_i|)$. Krachun--Petrov's conjecture.

$$\lim_{n\to\infty}\min_{A\subset\mathbb{R},\ |A|=n}\frac{|A+\lambda\cdot A|}{|A|}=H(\lambda).$$

This conjecture generalizes the sharp asymptotic lower bound known for $\lambda=\sqrt{2}$; the source notes that Krachun and Petrov prove the matching upper bound, while the general lower bound is the subject of the conjecture.