Changing the active tool in an ArcGIS Python Add-in Tool Palette?

I have an ArcGIS Python Add-in project with a tool palette that contains a dozen different tools. I would like to be able to change the 'active tool', i.e. the tool that is visible on the toolbar, when a button on the same toolbar is clicked.

The active tool changes when the user selects a tool from the dropdown palette, but I'm not sure if this behaviour can be replicated using Python.