Checker Deselect for single object which can consist from a few disconnected vertices parts (groups)

I have an object which consist from four parts of separate vertices and edges. I want the command [ checker deselect ] to affect all the vertices. Example:

It seems like a For loop is missing. and the action is done only on the first instance.

This tool works starting from the active element.
https://developer.blender.org/T73836

In case of islands of selected elements, there will be a random starting element.
So you will get different (shifted) patterns on the islands. This is probably not very useful.