I dont think thats one piece of A4 anymore... you had to cut it somewhere

I mean how heavy/thick the paper is? You have your regular 80 gram per square meter, some stuff in the middle and heavy 200 gram per square meter... beyond that its cardbaord.
It was 400 psi and 60cc of air. It flew out, tumbled, hit the target sideways and bounced back at me... almost hit me in the feet.
From the picture I can see the paper is pretty thin... if you got the nail far enough inside the cone and if it doesnt turn to confetti when you fire, I think it shouldnt tumble that much but it wont be very stable either.
Balancing diameter, nail size, cone length and cg of nail darts is some tricky stuff because it can only be done through trial and error. Or calculations if you're an aerospace engineer

Have you test fired it?