They almost certainly will, yes. Especially given that all the popular vaccines are spike protein vaccines only, which are relatively easy for SARS-2 to “learn” how to evade. Exposure to the whole virus provides a much broader set of epitopes for the immune system to learn.

I’d add that anyone who got vaccinated so that they can show proof of vaccination needs to understand that there’s no getting off the train. If you don’t get your booster you’ll be - gasp! - not fully vaccinated.