Let $(U_n)_n$ be an arbitrary sequence of open subsets of the unit disk $D(0,1)\subseteq \mathbb{R}^2$ s.t. $\sum_{n=0}^\infty \lambda(U_n)=\infty$ (where $\lambda$ is the Lebesgue measure). **Does there exist a sequence $(q_n)_n$ in $\mathbb{R}^2$ s.t. $D(0,1) \subseteq \bigcup_{n=0}^\infty (q_n+U_n)$?**

With the notation $q_n+U_n$, I mean $$q_n+U_n:=\{x\in \mathbb{R}^2|x-q_n\in U_n\}$$

EDIT: Fedor Petrov was quick to find an easy answer to this one and I'm forced to accept it. His method doesn't hold up though if I additionally demand that all the $U_n$ are convex. So, submissions with a take on such a related question are still welcome (although I'll not be able to reward your submission with a "accepted answer" badge)

UPDATE: Acting on popular request, I've reposted the revised question over here.