www.wysiwygwebbuilder.com

Is there a way to save the ToolBox/Object Manager panel width size once it's width is adjusted ?
Closing and opening WB resets it's width to the original size of the panel, not the adjusted size.
This is not the case with WB's right side & bottom panels. They remain the width they have been adjusted to.

The system should remember the last size.
Note that this standard functionality so I have no control over how this works.

Yes, the right and bottom panels do remember the last size, whether increased or decreased from the default.
However, the left panel will only remember the last size if the size is increased.
If one decreases the size (from the default size) it will not remember the smaller size.
In my case, on a 40" 4K screen, the ToolBox icon & text only take up 25% of the panel width.

Not a serious issue, but the reason I got the 40" 4K was for the screen real-estate. That's a lot of wasted real-estate.

It's possible that the registry key is corrupt.
In that case, this may be helpful:
viewtopic.php?f=10&t=22937

Yes, that was the solution. easy peasy
Thanks Pablo for creating the tool to reset.