No. For that, there would have to be registration rights specified in the contract. Tag-along, drag-along, etc., but there is none of that described.
If they file an S-1, I would expect that they would be registered with all the others, but that's probably not going to happen unless there is a move to the NASDAQ or NYSE, so who cares at that point. If they do a limited registration for a financing st some point, they almost certainly would not get registered.
It's possible to to get the restrictive legend removed by the TA under Rule 144 after 6 months if reporting is current, but because it requires a legal consent from the issuer's counsel, it's a practical impossibility for officers because the risk to the company is just too high to have officers running around selling unregistered shares to the public.
Maybe Truitt could sell his shares at some point if he's not involved, but Jay will likely be locked up like Squires and the other insiders until there is a sale of the company or uplisting.