Update: Okay, I finally found this tool in Firefox. Let’s say it did something. I'm not sure exactly what it was, other than opening several tabs quickly and then closing all at once. In the end, I was hoping this would restore icons from my huge backlog of bookmarks but as the description says: "...This extension is ONLY meant to simplify the process of opening every tab, so Firefox can load the fav icon - it does NOTHING else besides this..." Sad to say the tool loaded pages without any favourite icons appearing except for those I already had visited. Would be more of a great tool if it could also check missing or broken bookmark links and icons as well.

Does not do the work. It ran OK on 612 bookmarks and when done, all remain the same. Tried twice; 2nd time was after firefox restarted.

Update 1: Thanks for your reply and advice. Following the updated description and still, zero bookmarks get updated. After clearing all possible cache and history and all instances of favicons.sqlite could be found. After clearing all, all bookmarks does not have icon. After plugin run, still same: no icons. Same after FF restart. Sorry. Let me know if I can help with anything to debug. Using FF 121.0 (64-bit)

Update 2:
I have a little idea/suggestion: it seems as if the loading of the bookmarks runs too fast. In my testing, it takes 5-10 seconds for the icon to show up in Firefox's bookmarks toolbar. So maybe you can wait a longer time for the web site to make sure it was fully load? Or maybe add in the Settings a custom waiting time which I can manually set to slow down?

Edition Windows 10 Home
Version 22H2
Installed on ‎15-‎Jun-‎20
OS build 19045.3803
Experience Windows Feature Experience Pack 1000.19053.1000.0

Processor Intel(R) Pentium(R) CPU N4200 @ 1.10GHz 1.10 GHz
Installed RAM 4.00 GB (3.83 GB usable)
Device ID 9655FD22-A435-4D3C-B135-919A5A410E08
Product ID 00356-02037-68746-AAOEM
System type 64-bit operating system, x64-based processor
Pen and touch Touch support with 10 touch points