### 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 (p_addon.begins_with("res:")) {
		return addon_name_to_plugin.has(p_addon);
	}
	return addon_name_to_plugin.has("res:addons/" + p_addon + "/plugin.cfg");
}
void EditorNode::_remove_edited_scene(bool p_change_tab) {
	
	hide_unused_editors(SceneTreeDock::get_singleton());
	int new_index = editor_data.get_edited_scene();
	int old_index = new_index;
	if (new_index > 0) {
		new_index = new_index - 1;
	} else if (editor_data.get_edited_scene_count() > 1) {
		new_index = 1;
	} else {
		editor_data.add_edited_scene(-1);
		new_index = 1;
	}
	if (p_change_tab) {
		_set_current_scene(new_index);
	}
	editor_data.remove_scene(old_index);
	_update_title();
	scene_tabs->update_scene_tabs();
}
void EditorNode::_remove_scene(int index, bool p_change_tab) {