Possible, but unlikely.
The most widely-discussed plan for BU to start mining blocks larger than 1MB is the following ordered sequence:
1) achieve 75% mining dominance;
2) maintain that 75% or better dominance for a full difficulty retargeting period;
3) wait another difficulty retargeting period for others to upgrade to a cliant that does not fall down when faced with larger blocks;
4) only then fork.
With 75% dominance, and assuming 2MB blocks, many more transactions can be processed each day on the BU fork (~250,000/day to ~375,000/day), and fees would drop.
The legacy fork, at 25% mining capability (max) would see its transaction capability drop from ~250,000/day to ~62,500/day. Competition to get transactions included in the few blocks that do get mined will cause fees to skyrocket.
Miners on the BU fork will see BTC/day from subsidy drop by 25%, but miners on the legacy fork will see BTC/day drop by 75%. This will likely cause miners to further abandon the legacy fork, thereby slowing the legacy chain even further, and speeding the BU chain up even more.
This condition will persist for a full difficulty retargeting period, which will be lengthened on each chain. The BU chain would be slowed down for about 20 days, and the core chain would be slowed down for at least 8 weeks - proportionally more depending on number of miners who switch from legacy to BU.
And what would be the likely consequences?
The net result is likely that the legacy chain will likely die off for all but an obstinate 1%. Accordingly, BU would likely end up being thought of as 'the Bitcoin' at the end of this process.
The good news is that for any coin you hold in Bitcoin before the fork, you will have one Bitcoin on each fork after the split. Accordingly, while one chain atrophies to insignificance, you will still hold as many Bitcoin on the victorious chain.
If this all plays out as above, it is likely that the value of the BU fork -- within days of the fork -- will be though of as merely Bitcoin, and valued at only slightly less than Bitcoin before the fork. The legacy chain would be essentially worthless. And now, with the transaction throughput hard-cap removed, transactions/day will go up, fees will go down, Bitcoin will be able to fulfill use cases that have been driven away, and Bitcoin's price will climb rapidly.