Let $\alpha_1, \ldots, \alpha_n$ be algebraic numbers and let $p_1, \ldots, p_n$ be the corresponding minimal polynomials with integer coefficients.

Denote by $H$ the maximal magnituge among all coefficients of all $p_i$.

Assume that $\alpha_1 + \ldots + \alpha_n \not= 0$. Is it true that $|\alpha_1 + \ldots + \alpha_n | \ge {(CH)}^{-q(\text{deg}(p_1) +\ldots \text{deg}(p_n) )}$ for some fixed polynomial $q$ and constant $C$?

Warning: $C$ and $q$ should not depend on $\alpha_1, \ldots, \alpha_n$.

UPD1: it seem that this question is rather difficult. Let us consider a particular case: when all $\alpha_i$ are quadratic irrationality. Is it possible to prove the required result about the measure of its linear independence?

UPD2: Is it possible to prove the statement under some reasonable hypothesis like effective abc-conjecture or Vojta-conjecture?

UPD3: It is possible to prove that there exist some $C$ such that $|\alpha_1 + \alpha_2| \ge H^{-C(\deg(p_1) + \deg(p_2))}$

(if $\alpha_1 + \alpha_2 \not=0$)?