  The best known lower and upper bounds on the mixing time for the
random-to-random insertions shuffle are $(1/2-o(1))n\log n$ and $(2+o(1))n\log
n$. A long standing open problem is to prove that the mixing time exhibits a
cutoff. In particular, Diaconis conjectured that the cutoff occurs at $3/4n\log
n$. Our main result is a lower bound of $t_n = (3/4-o(1))n\log n$,
corresponding to this conjecture.
  Our method is based on analysis of the positions of cards yet-to-be-removed.
We show that for large $n$ and $t_n$ as above, there exists
$f(n)=\Theta(\sqrt{n\log n})$ such that, with high probability, under both the
measure induced by the shuffle and the stationary measure, the number of cards
within a certain distance from their initial position is $f(n)$ plus a lower
order term. However, under the induced measure, this lower order term is
strongly influenced by the number of cards yet-to-be-removed, and is of higher
order than for the stationary measure.
