The advisor's per-minute fee is displayed to you on multiple occasions.
1) The category pages & the advisor's profile page.
2) The "Pre-Session" page - The page you get to after hitting "Let's chat", where you press "Start Session" in order to start a session.
3) The session window itself.
The advisors can change their per-minute fee any time, until a session starts. The fee displayed to you prior to starting the session is static. This means that the change is not automatically reflected to you as while idle on a page. Only if you were to refresh, or navigate from the page, would you see the change.
Once you hit "Start Session", and the advisor is notified that you are contacting them, the advisor can no longer increase their per-minute fee.
After your session starts, once your advisor starts typing, the price can no longer be changed, higher nor lower. The advisor can only lower his session fee, before he starts typing
The per minute fee that you are charged by is the fee displayed once the advisor starts typing.
If you notice the fee has changed, you have an opportunity to ask your advisor to match the price you previously saw. However, if the advisor is already typing, they can't change their fee, you would have to end that session and start a new one.