Another important, poorly understood question is whether one can color rectangles with at most $O(omega(R))$ colors where $omega(R)$ is the size of a maximum clique in the intersection graph of a set of input rectangles $R$. Asplund and Gr\"{u}nbaum obtained an upper bound of $O(omega(R)^2)$ about 50 years ago, and the result has remained asymptotically best. This question is strongly related to the integrality gap of the canonical LP for MWISR. In this paper, we settle above three open problems in a relaxed model where we are allowed to shrink the rectangles by a tiny bit (rescaling them by a factor of $(1-\delta)$ for an arbitrarily small constant $\delta > 0$.) Namely, in this model, we show (i) a PTAS for MWISR and (ii) a coloring with $O(\omega(R))$ colors which implies a constant upper bound on the integrality gap of the canonical LP. For some applications of MWISR the possibility to shrink the rectangles has a natural, well-motivated meaning. Our results can be seen as an evidence that the shrinking model is a promising way to relax a geometric problem for the purpose of better algorithmic results.