[Bug - Edge Addons] Sidepanel API: Behavioral inconsistencies with Chrome implementation

Describe the bug

• Inconsistency 1: Stale Sidepanel Path on First Open

  • Chrome Behavior: When the sidepanel is opened on a tab and then opened on a subsequent tab with an updated path option, the sidepanel immediately displays the correct path for the current tab.
  • Edge Behavior: When opening the sidepanel on subsequent tabs with an updated path, Edge displays the stale path from the first tab where the sidepanel was opened. The path only updates correctly after closing and reopening the sidepanel

• Inconsistency 2: Sidepanel State Behavior When Switching Tabs

  • Chrome Behavior: When a sidepanel is opened on specific tabs, it remains open on those tabs even when navigating to/from tabs where the sidepanel was never opened.
  • Edge Behavior: When navigating to a tab where the sidepanel was never opened, Edge closes the sidepanel on other tabs as well, including tabs where it was previously opened.

• See https://github.com/pixiebrix/ms-edge-sidepanel-state/blob/main/README.md#summary

To Reproduce
Steps to reproduce the behavior:

1. Clone the https://github.com/pixiebrix/ms-edge-sidepanel-state repo
2. Follow the Setup Instructions from the readme
3. Follow the test cases from the readme

Expected behavior
See the results sections for each test

Screenshots

https://www.loom.com/share/25cc4f8b19a449f2b26c15c0982250e1

Desktop (please complete the following information):

• OS: MacOS (also reproducible on Windows 11)
• Browser: Edge
• Version: 142.0.3595.94

Additional context

• Potentially related to, but not the same as, #222