Swap Library function won't affect Icons

I got the following problem:

At a project we have three brands which will have all the same pages and components. So, my thought was to have a global library, without any branding and three different Libraries with brand specific styles, fonts, colors and components (like header or footer), etc…

Then I would add another file for each brand where I would display all the pages. For that I would copy the components or pages from the global library, paste them in the file and use the swap library function to apply the correct branding on the components.
To this point it all works fine.

But unfortunately, I noticed that Icons inside a component (instance from global library) won’t swap, even so they have the same name.
If I detach the component and try the swapping function again, the icon will swap to the right brand. I also added a video that might help.

Of course, this is not what I had in mind. I need the connection of the component to the global library, because if I change something inside a component I have only one place where I need to do it. If I detached, I would need to go into three libraries to change it.

Is there a way that the icons will swap also inside an instance?

Thank you