Let $\beta\in(1,2)$ and $\gamma\in(1,2)$ be Galois conjugates of height 1. That is, there exists a polynomial $p$ with coefficients $-1,0,1$ such that $p(\beta)=p(\gamma)=0$ (not necessarily minimal). 


Numerically, there appears to be an absolute constant $C>0$ such that $|\gamma-\beta|\ge C$. Is this true/known? If it is, what is the best known value for $C$? 


I've looked into some literature on root separation but couldn't find this claim. 



Positive real root separation
The polynomial


$$1 + x^n + x^{2n} - x^{3n} - x^{5n} - x^{6n} + x^{7n}$$


is irreducible and has two Galois conjugate roots $\beta\_n$ and $\gamma\_n$ in $(1,2)$ with


$$| \beta\_n - \gamma\_n| \sim \frac{\log(\beta\_1/\gamma\_1)}{n} \rightarrow 0.$$


