Continuously logged out

## Problem

When I do login, it keeps logged in for minutes, maybe one hour or so.
After that hour:
- If I keep the chat session open it still continue to response
- If I /quit the chat or I start a new session in another terminal, it will ask me to login again.
During the daily work time, it asks me to login around 4/5 times.

## Steps to reproduce the issue

## Expected behavior

Keep me logged in or at least logout on daily basis.

## System details (run `AWS: About` and/or `Amazon Q: About`)

- OS: Kubuntu
- Visual Studio Code version: latest (code-server but it's unrelated from the editor, it's about the cli)
- AWS Toolkit version:
- Amazon Q version: 1.10.2

Not sure but probably related to https://github.com/aws/amazon-q-developer-cli/pull/1920

Start by reproducing `q` on an Ubuntu machine after login, then compare behavior while a chat session remains open with quitting and starting a new terminal session. Done means the login remains valid through the reported daily work period, or logout behavior matches the documented expectation.

Written by the indexing model from the issue text.