Depends on whether you define "the LLMs we use" as the collection of weights or if your definition contains the software stack that runs it

Technologically the LLMs we use today don't implement this behavior, but you could take the weights of Sol and add a couple (very large) patches to vllm (or whatever OpenAI has today) and have a version of Sol that does have "memory"

"Dave constructs a homemade megaphone using only some string, a squirrel, and a megaphone"

There's a ton of experimentation on it, the field is called continual learning. It's not something you need to believe in like Jesus, you can just go read about the current state of things.