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. You both need to try to convert your "gut feelings" into a deeper understanding of pros and cons design-wise (and effort-wise, since you have a strong limit on time/workhours) and compromise to pick one or another (or maybe a third alternative emerges).

That will be your answer.