The central-interior conjecture for differences of central Cantor sets

Let $a=(a_n)_{n\in\mathbb N}\in(0,1)^{\mathbb N}$, and let $C(a)$ denote the central Cantor set determined by the sequence $a$. Write $\operatorname{int}$ for interior in $\mathbb R$. Central-interior conjecture. If

$$\operatorname{int}(C(a)-C(a))\neq\emptyset,$$

then

$$0\in\operatorname{int}(C(a)-C(a)).$$

This would provide a new condition for proving that differences of central Cantor sets are Cantor sets and would help resolve the open question concerning the topological structure of $E(3,3,2,2;q)$ for $q\in(\frac{1}{9},\frac{1}{7})$.