Property of Lebesgue measure in $\mathbb{R}^2$, part 2

Question

Let $I=A\times B,$ where $A,B\subset \mathbb{R}$ are closed sets of positive Lebesgue measure, and $E\subset \mathbb{R}^2,$ be a set of zero Lebesgue measure. Is it true that $$\overline{I\setminus E}=I?$$

When $I=[0,1]\times [0,1]$ the answer is true and one can find a solution in the following link. Property of Lebesgue measure in $\mathbb{R}^2$

Answer 1

No. Set $A=B=[0,1]\cup\{2\}$. Let $E$ be the singleton $\{(2,2)\}$. Then $\overline{I\setminus E}\neq I$.