Unless I am misunderstanding your question, the answer is embedded in the question itself.
The purpose of a good gray-scale card is to give you a standardized point of reference for what is true black, true white, and the intermediate shades of gray in between. So if you hold that card up to your subject, the card will be illuminated under the same lighting conditions as your subject… so it will experience the same color shifts that your local lighting conditions produce.
So, for example, if the white in your subject looks a bit yellow under incandescent lights, the true-white (value 10) on the card will look a bit yellow, too. But as long as your subject and the value on the card look the same, you know the color of the subject is truly "true white."