renaming 'Other Bookmarks' on IOS does not propagate changes to desktop - follow up to 7639

[GeetaSarvadnya]

Devices

Device A - Windows 1.7.x
Device B - Windows 1.8.x
Device C (mobile C) - iPhone 8 - 1.15(20.03.05.14)

Description:

Follow up to #7639 - specifically the "Rename" test plan from PR brave/brave-core#4404

Steps to Reproduce

1. Go to about:flag#brave-sync and enable sync on desktop A and desktop B
2. Sync desktop A, desktop B and then sync mobile C
3. Wait until each device has 3 devices on its device list
4. Add folder A in "Other Bookmarks" permanent node on desktop A
5. Desktop B should have folder A appears in “Other Bookmarks” permanent node
6. Mobile C should have folder A appears in “Other Bookmarks” folder
7. Add folder B after folder A on mobile C
8. Desktop A and desktop B should have folder B after folder A
9. Rename “Other Bookmarks” folder on mobile C
   --> Desktop A & B should have all children of "Other Bookmarks" moved to the renamed folder

Once above is working, should also confirm these steps from the original PR (they are also not working currently)

• Add folder D on desktop A in "Other Bookmarks" permanent node
• Confirm mobile C should have new "Other Bookmarks" folder created with folder D in it

Actual result:
renaming 'Other Bookmarks' on iPhone does not propagate changes to the desktop

Expected result:
Desktop A & B should have all children of "Other Bookmarks" moved to a renamed folder

Reproduces how often: [Easily reproduced, Intermittent Issue]

Always
Brave Version:
1.15(20.03.05.14)

Device details:

iPhone 8 - IOS 13.3.1
Website problems only:

• did you check with Brave Shields down? NA
• did you check in Safari/Firefox (WkWebView-based browsers)? NA

Additional Information

cc: @brave/legacy_qa @AlexeyBarabash @darkdh