### Tested versions

v4.2.stable.official [46dc27791]

### System information

Windows 11- Godot v4.2.stable.official [46dc27791] - Vulkan (Forward+)

### Issue description

When I switch picker mode the next time I open it should be the pick mode I selected last time but it's not

https://github.com/godotengine/godot/assets/46812028/522c8b55-14cf-4781-963f-a465a03c9432



### Steps to reproduce

Toggle a picker mode off and on again

### Minimal reproduction project (MRP)

N/A
if (singleton->cmdline_export_mode) {
		print_line("\t" + p_task + ": step " + itos(p_step) + ": " + p_state);
		return false;
	} else if (singleton->progress_dialog) {
		return singleton->progress_dialog->task_step(p_task, p_state, p_step, p_force_refresh);
	} else {
		return false;
	}
}
void EditorNode::progress_end_task(const String &p_task) {
	if (singleton->cmdline_export_mode) {
		print_line(p_task + ": end");
	} else if (singleton->progress_dialog) {
		singleton->progress_dialog->end_task(p_task);
	}
}
void EditorNode::progress_add_task_bg(const String &p_task, const String &p_label, int p_steps) {
	singleton->progress_hb->add_task(p_task, p_label, p_steps);
}
void EditorNode::progress_task_step_bg(const String &p_task, int p_step) {
	singleton->progress_hb->task_step(p_task, p_step);
}
void EditorNode::progress_end_task_bg(const String &p_task) {
	singleton->progress_hb->end_task(p_task);
}
String EditorNode::_get_system_info() const {