Well to be 100% sure the all touch each other, I can only make a 6 stick formation like this.
(I made the top two ones "transparent" in order to show that they all actually touch other, and yes they rest on top of the others).
EDIT - Scrap this.
If the sticks have the same length, no way the central one can fit there and let the formation actually work.
Maybe we can remove it, and we are left with a 5 stick solution.