### 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
ProjectSettings *ps = ProjectSettings::get_singleton();
	PackedStringArray enabled_plugins = ps->get("editor_plugins/enabled");
	for (int i = 0; i < enabled_plugins.size(); ++i) {
		if (enabled_plugins.get(i) == p_name) {
			enabled_plugins.remove_at(i);
			break;
		}
	}
	ps->set("editor_plugins/enabled", enabled_plugins);
}
void EditorNode::_plugin_over_edit(EditorPlugin *p_plugin, Object *p_object) {
	if (p_object) {
		editor_plugins_over->add_plugin(p_plugin);
		p_plugin->make_visible(true);
		p_plugin->edit(p_object);
	} else {
		editor_plugins_over->remove_plugin(p_plugin);
		p_plugin->make_visible(false);
		p_plugin->edit(nullptr);
	}
}
void EditorNode::_plugin_over_self_own(EditorPlugin *p_plugin) {
	active_plugins[p_plugin->get_instance_id()].insert(p_plugin);
}
void EditorNode::_resources_changed(const Vector<String> &p_resources) {