I have never seen a bird do this...but I have never had a young conure, so I am hoping others chime in.
Does he ever sleep or climb on his perch?
Where is he laying down? On the cage floor?
Is he weaned and how old was he when you got him?
I Googled it and there seem to be some bird-owners on various sites saying that their birds did this (but without speaking to them directly, it's impossible to know whether or not their birds were healthy). For many older parrots (still young, but fully weaned etc), sleeping on the floor or difficulty perching for sustained periods can be a sign of illness, but given your bird's age and species, I am not sure....again-- never owned a conure...It could be a quirky conure baby thing..your vet could tell you better than I.
I would get in touch with a vet just because your bird would need to go no matter what in the near future (just for preventative care etc).
If he is eating and acting fine otherwise, that is a good sign, but he needs to get checked up anyway.... Birds are masters at hiding illness (without any obvious symptoms). That is why yearly blood-work/wellness-checkups are so important for all birds. A bird that seems to be 200% healthy still needs to go to the vet at least 1x each year for blood-work etc.
Again, this behavior seems very strange to me, but I haven't ever had a conure, so I am basing it on what I have seen in my birds/familiar birds over the years...I am by no means a conure expert...plus, your bird is very young