Under what conditions does a non measurable dense set necessarily intersect every closed uncountable set?

Question

Let $D\subset \Bbb{R}$ be a dense set.

Question: Does $D$ intersect every closed uncountable sets?

The answer is negative. As the Cantor set $\mathcal{C}$ is n.w.dense , it's exterior $\Bbb{R}\setminus \overline{\mathcal{C}}=\Bbb{R}\setminus {\mathcal{C}}$ is dense in $\Bbb{R}$ but it doesn't intersect the closed uncountable set $\mathcal{C}$.

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Here we want to focus on non measurable dense sets.

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Under what assumption on the the dense set makes the desirable conclusion true?

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$A\subset \Bbb{R}$ is called saturated non measurable set iff$$m_i(A) =0=m_i(A^c)$$

Where $m_i(A) $ is the inner measure of $A$ and $A^c=\Bbb{R}\setminus A$

For an example Bernstein set is a saturated non measurable set.

Claim: $A\subset \Bbb{R}$ be a saturated non measurable set then $A$ is dense.

Lemma: Let $A\subset \Bbb{R}$ be such that $m_i(A^c) =0$.Then $\forall E\in\mathcal{L}(\Bbb{R}) $ with $m(E) >0$ we have $A\cap E\neq \emptyset$

Proof (of the Lemma): Assume the contrary that is $A\cap E=\emptyset$.

Then $E\subset A^c$ implies $m_i(A^c)\ge m_i(E)=m(E) >0$.

This contradict our assumption that $m_i(A^c) =0$

Proof (of the claim) : Any non empty open interval $(a, b) $ is a measurable set with positive measure. Hence from the above Lemmma , it is clear that $A\cap (a, b) \neq \emptyset$.

Hence $A$ is dense in $\Bbb{R}$

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Let $A\subset \Bbb{R}$ be a saturated non measurable set.

Question: Does $A$ intersect every closed uncountable sets?

Bernstein set is an example of a saturated non measurable set and Bernstein set intersect every closed uncountable sets.

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Conjecture : A saturated non measurable set $D\subset \Bbb{R}$ intersect every closed uncountable sets.

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If the conjecture can be proved to be true then it will help me to solve this question easily.

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UPDATE:

The conjecture is false.

Let $D\subset \Bbb{R}$ be saturated non measurable set and $F$ be a closed uncountable set of measure $0$.

Then $A=D\setminus F$ is a saturated non measurable set as $A\subset D$ and $A^c\subset F$ and $m_i(D) =0=m_i(F) =m(F) $.

But clearly $A$ doesn't intersect $F$.

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1. Dense set may not intersect every closed uncountable sets.

2. Saturated non measurable set may not intersect every closed uncountable sets.

Question:Under what conditions does a non measurable dense set necessarily intersect every closed uncountable set?

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