Euclidean geometry – covering the disk with a family of infinite dimensions – the convex continuation

To let $ (U_n) _n $ be an arbitrary sequence of open convex 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). Is there a sequence? $ (q_n) _n $ in the $ 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
$$ q_n + U_n: = {x in mathbb {R} ^ 2 | x-q_n in U_n } $$

This question is very similar to this one, but I was asked in the comments to ask anyway.