Set prompt to aistudio via url query parameter

Similar to what gemini.google.com has on google chrome when using search engine → site search → @gemini search pattern, to automatically send a prompt from search bar to the web app, which basically sends the prompt via x-omnibox-gemini http request header

curl 'https://gemini.google.com/prompt' \
  -H 'x-omnibox-gemini: <PROMPT>' \
  -b '...' \
  -H 'user-agent: Mozilla/5.0 (X11; Linux x86_64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/134.0.0.0 Safari/537.36' \
  -H 'x-browser-validation: <REDACTED>' \
  -H 'x-client-data: <REDACTED>' \
  ...

Could be something like → https://aistudio.google.com/app/prompts/new_chat?model=gemini-2.5-pro-exp-03-25&prompt=<PROMPT>

I know this is kind of sets some specific constraints in terms of the types of prompts that would be supported supports via url on search bar, but this could be very useful for simple queries.

Kind of similar to what chatgpt has with https://chatgpt.com/?hints=search&q=<PROMPT>

I know this could fits better in gemini.google.com but in aistudio.google.com there are more knobs that can be used that are not available in gemini.google.com like temperature, grounding, etc.

This could be awesome to use in combination with Set temperature value on aistudio via url query parameter and Enable "Grounding with Google Search" switch on aistudio via url query parameter