Not really. Here is what will happen:
1 - The underwriter will determine which holders it wants to have sign a lock-up agreement.
2 - If I am asked to sign a lock-up agreement the underwriter will have a lot of leverage to make me sign the damn piece of paper. I've literally seen them refuse to take companies public until all individuals and entities they have selected to sign the lockup agree to sign. Who they ask to sign is based on how they are affiliated to the company. And it is typically a very select group of people.
3 - When I sign the lockup agreement I'll be contractually obligated to abide by its terms.
4 - If I'm not asked to sign the lockup agreement, I'll be free to sell my shares into the open market - subject to any volume restrictions prescribed by law that may apply - and no one will care.
Never mind the fact that you are presuming how I behave in one private placement turned IPO somehow impacts whether other private businesses are willing to take my money. Which is way off base. The odds I'm investing through the same group of people are ridiculously low. And the odds they have a business that (a) goes public as opposed to realizing another exit event, (b) uses the same underwriter, and (c ) have an underwriter that wants me to sign a lockup in the first place are all really low. The person I'd piss off by refusing to sign a lockup is the underwriter - and the underwriter doesn't have some magic ability to put me on some global private placement blacklist.