Something doesn't seem right here. Let's ignore the 'coefficient of friction' and just make the sticks out of caramel: if they even touch they glom together. Nevertheless, everything is always flat!
If sticks ever glom, there's a pair of sticks that glom first. However, if you think about a pair of sticks falling in this way (I'm assuming $\epsilon$ is small, so all sticks make the same angle to the horizontal at any moment before the first glom), it's clear that they only actually make contact with each other at the instant they hit the ground. (The only other possibility is having the tips collide, which almost surely never happens.)
Perhaps you can make the sticks thicker and have something happen, but that seems extremely complicated.
