ADDITIONAL SYSTEM INFORMATION :
The bug is still happening in JavaFx 14 and also JavaFx 15-ea+3

A DESCRIPTION OF THE PROBLEM :
I opened https://bugs.java.com/bugdatabase/view_bug.do?bug_id=JDK-8237602 which is now marked as fixed because the related issue JDK-8222457 is fixed.

However there is an important difference between these 2 issues: the JDK-8222457 reports a problem when changing the order of the tabs but not the number of tabs and this issue is fixed indeed. But JDK-8237602 reports a problem when changing both the tabs order and the number of tabs and this issue is not fixed.

STEPS TO FOLLOW TO REPRODUCE THE PROBLEM :
See JDK-8237602

EXPECTED VERSUS ACTUAL BEHAVIOR :
EXPECTED -
See JDK-8237602. Now it works on second button click but it's not completely fixed as it should work on first button click. 
ACTUAL -
See JDK-8237602.

When pressing the button on first time, the UI still displays the tabs in this order: Tab 3, Tab4, Tab 1, Tab 2 instead of Tab 1, Tab 2, Tab 3 & Tab 4 because the button action change both the order and the number of tabs (from 2 to 4). Then if you click a second time on the button the tabs are displayed in the correct order because this time only the order have changed (the number of tabs stays at 4). But the correct behaviour should be that it works on the first button click.

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See JDK-8237602
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CUSTOMER SUBMITTED WORKAROUND :
See JDK-8237602

FREQUENCY : always