With the implementation of the additional tabs being done via a new chest at the top of the stash space screen, why not create an additional chest that is shaded in green. When you complete a set dungeon it gives you another set tab that you can place only the set items for that set into. This is done for all four sets for each class. The tab is class specific and is shared across all characters of that class. If you wanted to take it a step further you could add the outfitter feature to each tab and when you complete all set dungeons you get a 5th tab under the set tab to place any build you want into it.
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With the implementation of the additional tabs being done via a new chest at the top of the stash space screen, why not create an additional chest that is shaded in green. When you complete a set dungeon it gives you another set tab that you can place only the set items for that set into. This is done for all four sets for each class. The tab is class specific and is shared across all characters of that class. If you wanted to take it a step further you could add the outfitter feature to each tab and when you complete all set dungeons you get a 5th tab under the set tab to place any build you want into it.