When you’re working with a client, they’ve hired you to fulfill a goal. To this end, you want them to have faith in your skills, so that they’re more inclined to listen to your recommendations and research. Thus, if they catch you looking like you're dead tired and running on little sleep during a video conference, your client might take it to mean that you're generally unenthused and/or don't care about their brainchild. Regardless of how you actually feel.
And you don’t want to insult your client’s brainchild.
Thus, while your actual work and research on a project is important, how you present that info to your client can make all the difference. So smile, be well dressed, and hide your tiredness until you get off that video call.