It would be nice if there is a way to customize QToolButton extended popup menu

The extended Popup menu shown after clicking right arrow button (This button gets added if you resize Tool Bar to very small)goes away the moment mouse moves out of the drop-down popup area.

It would be nice if it stays as long mouse is not clicked either inside or outside (like context menu) and further if there is way to get popup menu object so that it can be further customization for styling.