Erdős–Komornik conjecture on non-Pisot beta-expansions

Let $1<\beta<\dfrac{1+\sqrt{5}}{2}$, and let $L^{1}(\beta)$ denote the Lebesgue measure of the set of points in $[0,(\beta-1)^{-1}]$ that have a universal expansion in base $\beta$. Erdős–Komornik conjecture. For any non-Pisot $\beta\in\left(1,\dfrac{1+\sqrt{5}}{2}\right)$, $L^{1}(\beta)=0$. The conjecture concerns the size of the set of points with universal expansions; the supplied text presents it as an old conjecture connected to the Hausdorff dimension of the complementary set, and gives no resolution.