Almost quadratic gap between partition complexity and query/communication complexity

We show nearly quadratic separations between two pairs of complexity measures: 1. We show that there is a Boolean function $f$ with $D(f)=\Omega((D^{{sc}(f)){2-o(1)})$} where $D(f)$ is the deterministic query complexity of $f$ and $D^{sc}$ is the subcube partition complexity of $f$; 2. As a consequence, we obtain that there is a communication task $f(x, y)$ such that $D^{{cc}(f)=\Omega(\log{2-o(1)}\chi(f))$} where $D^{cc}(f)$ is the deterministic 2-party communication complexity of $f$ (in the standard 2-party model of communication) and $\chi(f)$ is the partition number of $f$. Both of those separations are nearly optimal: it is well known that $D(f)=O((D^{{sc}(f)){2})$} and $D^{{cc}(f)=O(\log2\chi(f))$.}