Shard execution isn't necessary for migrating to proof-of-stake. The PoS migration just links a single EVM to the beacon chain; that's enough to remove PoW, i.e. the main concern in this discussion about GPUs.

Around the same time is the migration to data sharding, which multiplies data storage by 23X initially. That's without code execution on the shards.

Zkrollups in production now do the execution off-chain. On chain, you store compressed state data and a compact proof that everything executed correctly. That's good for a couple thousand tx/sec today, and 23X that after data sharding. Given that, Vitalik has argued that shard execution may not even be necessary; it wouldn't add nearly so much transaction capacity as we'd have already.

Here are the papers on how Ethereum consensus actually works:

https://arxiv.org/abs/1710.09437

https://arxiv.org/abs/2003.03052

And here's more detail on rollups:

https://vitalik.ca/general/2021/01/05/rollup.html