The weak converse of Zeckendorf's theorem for measurable Zeckendorf subsets

Let

$$be a finite list of positive integers, and let$$

be the unit interval. A **Lebesgue measurable

$$-Zeckendorf \subset** of$$

is a subset satisfying the

$$-Zeckendorf condition described in the source. **Weak converse of Zeckendorf's theorem.** Given a finite list of positive integers$$

, each Lebesgue measurable

$$-Zeckendorf \subset of$$

that is not an open interval has measure zero.

This is a measure-theoretic converse asserting that the only non-null measurable subsets with the relevant Zeckendorf property are open intervals. The supplied text does not provide evidence resolving the conjecture.