It "forgets" that it triggered an auto-update

Describe the bug
If "Apply updates automatically" is enabled, and an auto-update has been triggered, Update Manager doesn't notify the user that an update is in progress. And when the user wants to explicitly allow an update it says "another process is using apt" (or something along those lines), which is correct, but very unhelpful, because the UM was the one responsible for the lock!

Screenshots
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(I'll add them later, when I have the opportunity)

To Reproduce
Steps to reproduce the behavior:

1. Go to the "Update Manager" app
2. Click on "Install Updates"
3. See software lock error

Expected behavior
It should show a message that an update is already in progress, preferably showing the well-known dialog-window with progress bar.

Distribution:

• Linux Mint
• LMDE
• Other (please specify)

Software version:
5.9.7

Logs:

01.18@13:36 ++ Launching Update Manager
01.18@13:36 ++ Changes to the package cache detected, triggering refresh
01.18@13:36 ++ Starting refresh (local only)
01.18@13:36 ++ Initial refresh will happen in 0 day(s), 0 hour(s) and 10 minute(s)
01.18@13:36 ++ Tracker: Mintupdate update button was pressed recently: 1 days ago
01.18@13:36 ++ Found 3 software updates
01.18@13:36 ++ Refresh finished
01.18@13:40 ++ Starting refresh (local only)
01.18@13:40 ++ Found 3 software updates
01.18@13:40 ++ Refresh finished
01.18@13:46 ++ Update Manager is in tray mode, performing initial refresh
01.18@13:46 ++ Package management system locked by another process, retrying in 60s
01.18@13:47 ++ Package management system locked by another process, retrying in 60s
01.18@13:48 ++ Package management system locked by another process, retrying in 60s
01.18@13:49 ++ Package management system locked by another process, retrying in 60s
01.18@13:50 ++ Package management system locked by another process, retrying in 60s
01.18@13:51 ++ Package management system locked by another process, retrying in 60s
01.18@13:52 ++ Package management system locked by another process, retrying in 60s
01.18@13:53 ++ Package management system locked by another process, retrying in 60s
01.18@13:54 ++ Package management system locked by another process, retrying in 60s
01.18@13:55 ++ Package management system locked by another process, retrying in 60s
01.18@13:56 ++ Package management system locked by another process, retrying in 60s
01.18@13:57 ++ Package management system locked by another process, retrying in 60s
01.18@13:58 ++ Package management system locked by another process, retrying in 60s
01.18@13:59 ++ Package management system locked by another process, retrying in 60s
01.18@14:00 ++ Package management system locked by another process, retrying in 60s
01.18@14:01 ++ Package management system locked by another process, retrying in 60s
01.18@14:02 ++ Package management system locked by another process, retrying in 60s
01.18@14:03 ++ Package management system locked by another process, retrying in 60s
01.18@14:04 ++ Package management system locked by another process, retrying in 60s
01.18@14:05 ++ Package management system locked by another process, retrying in 60s
01.18@14:06 ++ Package management system locked by another process, retrying in 60s
01.18@14:07 ++ Package management system locked by another process, retrying in 60s
01.18@14:08 ++ Package management system locked by another process, retrying in 60s
01.18@14:09 ++ Package management system locked by another process, retrying in 60s
01.18@14:10 ++ Package management system locked by another process, retrying in 60s
01.18@14:11 ++ Package management system locked by another process, retrying in 60s
01.18@14:12 ++ Package management system locked by another process, retrying in 60s
01.18@14:13 ++ Package management system locked by another process, retrying in 60s
01.18@14:14 ++ Package management system locked by another process, retrying in 60s
01.18@14:15 ++ Package management system locked by another process, retrying in 60s
01.18@14:16 ++ Package management system locked by another process, retrying in 60s
01.18@14:17 ++ Package management system locked by another process, retrying in 60s
01.18@14:18 ++ Package management system locked by another process, retrying in 60s
01.18@14:19 ++ Package management system locked by another process, retrying in 60s
01.18@14:20 ++ Package management system locked by another process, retrying in 60s
01.18@14:21 ++ Package management system locked by another process, retrying in 60s
01.18@14:22 ++ Package management system locked by another process, retrying in 60s
01.18@14:23 ++ Package management system locked by another process, retrying in 60s
01.18@14:24 ++ Changes to the package cache detected, triggering refresh
01.18@14:24 ++ Starting refresh (retrieving lists of updates from remote servers)
01.18@14:24 ++ Refreshing available Cinnamon updates from the server
01.18@14:24 ++ Refreshing available Flatpak updates
01.18@14:24 ++ System is up to date
01.18@14:24 ++ Refresh finished
01.18@14:24 ++ Recurring refresh will happen in 0 day(s), 1 hour(s) and 59 minute(s)

Additional context
None (for now, AFAIK)

Start by reproducing the Update Manager workflow with “Apply updates automatically” enabled, then click “Install Updates” while the automatic update is in progress. Trace the auto-update and package-lock handling from the Update Manager entry points mentioned in the issue. Done means the UI identifies its own update as already in progress and presents an appropriate progress indication instead of an unhelpful lock error.