Hide cursor for application layout using only buttons

I am building a small TUI application with python prompt toolkit, it is not using a prompt.
As there is nothing to input, I would like to hide the cursor. There are just a couple of buttons in a full-screen text application.

I tried to use application.output.hide_cursor() before application.run(), however that seems to be ignored. As I have no input, I can also not wrap the whole thing into a Window-object and set its always_hide_cursor property to True nor access the underlying Screen objects which seem to have an effect if the cursor get switched on in rendering or not.

Is there any way I can accomplish to disable the cursor in my case?

Start by tracing how application.output.hide_cursor() behaves across application.run(), then inspect the Window always_hide_cursor property and the underlying Screen rendering mentioned in the report. Determine how a full-screen button-only application can keep the cursor hidden, and verify that the behavior works without a prompt or input window.