Avoidable-set lower-density conjecture

Let $U$ be an avoidable set of integers, and suppose that $U$ contains infinitely many even numbers. Write $LD(U)$ for its lower logarithmic density. Lower-density conjecture.

$$LD(U)\leq\frac{\sqrt{5}-1}{2}.$$

The preceding theorem proves the analogous bound for $ELD(U)$, while the conjecture asks for the same upper bound for $LD(U)$. The source gives no resolution of this stronger assertion.