Do the fine tuned models bake in anything from system/user?

It seems like you can only assign weights to assistant messages. I am wondering, if I did something like:

system: today’s date is 2024-08-07

user: what is todays date

assistant: its 2024-08-07

If I finetuned the above, does the AI learn to always say it is 2024-08-07 when asked the same question on any day, because the finetuning data included a date in a dynamic system prompt that just happened to be the day finetuning was done?

Thanks!