  Given any set of points $S$ in the unit square that contains the origin, does
a set of axis aligned rectangles, one for each point in $S$, exist, such that
each of them has a point in $S$ as its lower-left corner, they are pairwise
interior disjoint, and the total area that they cover is at least 1/2? This
question is also known as Freedman's conjecture (conjecturing that such a set
of rectangles does exist) and has been open since Allen Freedman posed it in
1969. In this paper, we improve the best known lower bound on the total area
that can be covered from 0.09121 to 0.1039. Although this step is small, we
introduce new insights that push the limits of this analysis.
  Our lower bound uses a greedy algorithm with a particular order of the points
in $S$. Therefore, it also implies that this greedy algorithm achieves an
approximation ratio of 0.1039. We complement the result with an upper bound of
3/4 on the approximation ratio for a natural class of greedy algorithms that
includes the one that achieves the lower bound.
