Let $A,B$ be two homeomorphic topological subspaces of $\mathbb{R}^3$ such that their complements $\mathbb{R}^3 - A, \mathbb{R}^3 - B$ are not homeomorphic to each other. Must $A \cong B$ contain a homeomorphic image of the Cantor set?

(It is known that there are homeomorphic images $A,B$ of the Cantor set such that $\mathbb{R}^3 - A, \mathbb{R}^3 - B$ are not homeomorphic, see e.g. https://www.sciencedirect.com/science/article/pii/016686418690060X)

UPDATE 1:

The answer is no, as Wojowu's answer shows. This leads to
**Question 2:** Must $A \cong B$ contain a homeomorphic image of $\mathbb{Q}$?

UPDATE 2:

After Wojowu's answer, the interesting question remaining is

**Question 3:** Let $A,B$ be two closed, countable, topological subspaces of $\mathbb{R}^3$ homeomorphic to each other. Must their complements $\mathbb{R}^3 - A, \mathbb{R}^3 - B$ be homeomorphic to each other?