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.