This is opinionated question. It can be fun either way and boring as well. Only you (together) can decide what you like best and have a better chance of implementing well.
Start playtesting!
It does not have to be fancy - from what I can tell, you can try starting with an A5 piece of graph paper and play with your partner imposing 10-30 players with a set of color pencils. Or use an online Excel, or playtest in some online grid-cell 2D game like Factorio. You can even do it solo. Try it with static square size and try it with growing square sizes - see how it behaves:
- with growing player count
- dormant squares
- how world size restricts you
- how game dynamic feels
- how players get in the way of each other
- and how this can be resolved (e.g. player with 10x10 square level wants to capture 10x10 square area occupied by 100 different 1x1 players)
- how it scales up for 10, 100, 1k players
- most important - does it feel fun?
- etc.
Ask your partner about their concerns and scenarios - what plays well and what are the problem cases.
That will be your answer. And hopefully this helps not only with design, but also brings you back to common grounds.