Pinch's conjecture on $a$-convexity

For $a$ in $\mathbb{R}$, a subset $V$ contained in ${\mathbb{R}}^n$ is said to be $a$-convex if $x,y\in V⟹ax+(1-a)y\in V$. According to Pinch [Math. Proc. Cambridge Philos. Soc., 1985], the $a$-convex hull of $V$ is the intersection of all of the $a$-convex subsets of ${\mathbb{R}}^n$ that contain $V$, and Pinch also defines $D(a)$ as the $a$-convex hull of $\{0,1\}$ in ${\mathbb{R}}^1$. Pinch conjectured that if $a$ is a totally real algebraic integer and $D(a)$ has no limit points, then every algebraic conjugate of $a$ other than $a$ is in $(0,1)$. We succeed in proving this conjecture, which seems to have remained open.