When I place a dropper facing down into a chest it will not put the item in the chest. But it do work for other directions.

Image 1: The setup. The left dropper/chest will work, the right will not work.
Image 2: Just pressed the button, and you can see the item flying around.
Image 3: No item in the right chest (obvious need to remove dropper, to check)
Image 4: Item in the left chest, working as expected.