Let $n>0$, and let $p,q\in (0,1)$ such that $p<q$.

Is there a hyperrectangle $H$ that satisfies the following:

- $\forall i\in{1,\dots,n}:\\ H\supset \prod_{j=1,\dots,n} \begin{cases} [p,q], &\text{if j=i}\\ [0,1], &\text{else} \end{cases}$
- $\{0,1\}^n\cap H=\emptyset$

I tried many examples by trial and error, so I was able to conclude that there exists such a hyperrectangle (or actually, a square) in 2D, but I couldn't find such a hyperrectangle in 3D or in higher dimensions, or to disprove its existence.

Any help would be appreciated.