My 5 month old Shar Pei is not very social with me and my vet thinks I am going to have problems with him when he gets older. What do you think?

I have question on whether the pup I have may be at risk of being a fear biter. The dog is a five-month-old Shar Pei male, that we have had since he was 16 weeks. The mother was outgoing, father was more standoffish, but friendly to strangers at the dog show with the owner present. The pup was/is shy, and it has taken about 6 weeks since he has been with us to approach us with a wagging tale, but still reluctant to be petting on the head and neck (backs off), but more agreeable for pets under the neck and chest (less dominant approach). He is friendly to other dogs he meets (even if they are not friendly to him), and walks on a lead without fear of traffic noise. He will pull on the lead if he passes to close to strangers.

I have taken him to the vets, and to friends who have children. I don?t let them gang up on him, and he will sit quietly in their living room with the lead still on him if they are a few feet away. One child next to him was OK for a pet, no snapping. When I got him six weeks ago, you would never get him in a house or near a group of kids, and would snap at them then.

When I stopped by my neighbors with three adults, and they approached them, he pulled on the lead and snapped, but then settled down, and would allow one of them to pet him.

When delivery people come, he will bark for a few minutes, and then walk around the delivery trailer, and without much fuss, but keep a few feet away from them.

The in-laws came for a visit on the weekend, he was reluctant to go into the house when they were there, but after two days would follow behind them and sniff them with his tail wagging (but not allow any pets).

The vet has some concerns that as he gets older, he could still be a problem, as most dogs 5 months old are a lot more social, puppies are normally everybody?s friend. He is not really that social with me. I would not let him loose on an open field, not sure if I would get him back. I can still play with balls and stuffed toys on a string with him.

When he comes in at night, he prefers to sleep next to an open door. I will close it later, and then he settles down somewhere else in the house.

The breeder will still take him back up to the age of 6 months. Is this dog going to have a chance of being mentally sound and safe even with training?