[BUG] Q_TERM=1.7.2 even after upgrade/uninstall/restarting session

### Checks

- [x] I have searched [github.com/aws/amazon-q-developer-cli/issues](https://github.com/aws/amazon-q-developer-cli/issues?q=) and there are no duplicates of my issue
- [x] I have run `q doctor` in the affected terminal session
- [x] I have run `q restart` and replicated the issue again

### Operating system

Linux 6.9.3-76060903-generic - Pop!_OS 22.04 LTS

### Expected behaviour

After upgrading Q CLI to 1.7.3 it should stop stating that q is still on version 1.7.2

### Actual behaviour

When starting a new bash shell:
```
A new version of q is available: 1.7.3
Run q update to update to the new version
```

`Q_TERM` is still set to 1.7.2

### Steps to reproduce

```bash
sudo apt purge amazon-q
```

- confirm that .bashrc and .profile are clean

restart shell
```bash
curl --proto '=https' --tlsv1.2 -sSf https://desktop-release.q.us-east-1.amazonaws.com/latest/amazon-q.deb -o amazon-q.deb
sudo dpkg -i amazon-q.deb
sudo apt-get install -f
```

restart shell
```bash
q integrations install dotfiles
```

restart shell
```bash
q doctor --verbose
```

```
✘ Amazon Q terminal integrations: This terminal is not running with the latest integration, please restart your terminal

Q_TERM=1.7.2

✘ Doctor found errors. Please fix them and try again.

If you are not sure how to fix it, please open an issue with q issue to let us know!
```

### Environment

```yaml
[q-details]
version = "1.7.3"
hash = "3e4ae79d371315e80ddac772b43fff2cba314104"
date = "2025-04-10T05:51:23.996702Z (9d ago)"
variant = "full"

[system-info]
chip = "Intel(R) Core(TM) i7-8750H CPU @ 2.20GHz"
total-cores = 6
memory = "30.99 GB"

[system-info.os.linux]
kernel_version = "6.9.3-76060903-generic"
id = "pop"
name = "Pop!_OS"
pretty_name = "Pop!_OS 22.04 LTS"
version_id = "22.04"
version = "22.04 LTS"

[environment]
cwd = "/tmp"
cli-path = "/tmp"
os = "Linux"
shell-path = "/usr/bin/bash"
shell-version = "5.1.16"
terminal = "Kitty"
install-method = "unknown"

[env-vars]
DISPLAY = ":1"
GTK_IM_MODULE = "ibus"
PATH = "/opt/homebrew/bin:/usr/local/sbin:/home/USER/.local/bin:/usr/local/go/bin:/home/USER/go/bin:/home/USER/.local/share/pnpm:/home/USER/.local/bin:/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/games:/usr/local/games:/snap/bin:/home/USER/.local/bin:/home/USER/.fzf/bin"
QTERM_SESSION_ID = "7eeb276ccfe74c5d99d7acd1fdb96bb7"
QT_IM_MODULE = "ibus"
Q_SET_PARENT_CHECK = "1"
Q_TERM = "1.7.2"
SHELL = "/bin/bash"
TERM = "xterm-kitty"
XDG_CURRENT_DESKTOP = "pop:GNOME"
XDG_SESSION_DESKTOP = "pop"
XDG_SESSION_TYPE = "x11"
XMODIFIERS = "@im=ibus"
```