Hah no way. The poor LLM has no privacy to your prying eyes. I kinda like the 'reasoning' text it provides in general. It makes prompt engineering way more convenient.
Prompted appropriately of course it was happy to divulge ways to bypass it. I still haven't spent significant effort to extract the system prompt yet since running 32b or 70b is very very slow on my desktop. I should try with one of the smaller models.