Monomial ideals with tiny squares

Let $I\subset K[x,y]$ be a monomial ideal. How small can $\mu (I^2)$ be in terms of $\mu (I)$? It has been expected that the inequality $\mu (I^2)>\mu (I)$ should hold whenever $\mu (I)\ge 2$. Here we disprove this expectation and provide a somewhat surprising answer to the above question.