Let $P=A_1\times A_2,$ where $A_1,A_2\subset \mathbb{R}$ are set of positive Lebesgue measure, and $Z\subset \mathbb{R}^2,$ be a set of zero Lebesgue measure. Can we always find positive Lebesgue measure sets $B_1,B_2\subset \mathbb{R}$ such that $$B_1\times B_2 \subset \overline{P\setminus Z}?$$ What extra conditions ensure that the above is true?(I can show that the above is true if $P\setminus \overline{Z}$ is of positive measure then the above is true)

In this question https://math.stackexchange.com/q/3767758/641816, it was shown that the result is true if $A_1=A_2=[0,1]$.

**This is my attempt:** Since $A_1,A_2$ are positive Lebesgue measure set we can find $a_1\in A_1, a_2\in A_2$ such that for any $r>0$ we have $B(a_1,r)\cap A_1, B(a_2,r)\cap A_2$ are sets of positive measure(in fact this phenomenon is true for almost every $a_1\in A_1,a_2\in A_2$).
Consider $$B_1^r=\overline{B(a_1,r)\cap A_1},\quad B_2^r=\overline{B(a_2,r)\cap A_2}$$
Then I think somehow one can show that there exits some $s,t>0$ such that $$B_1^s\times B_2^t\subset \overline{P\setminus Z}.$$