I am about to enter a 3-year PhD. Neither my bachelor’s nor my master’s were research-oriented nor very typical for my field, and I had to find a potential supervisor on my own. Due to my background, I also do not have a good grasp of academia’s unwritten rules.
Last year, I fortuitously met a rock star professor, one of the top 5 most cited scholars in quite a large field. He got interested in my project, gave me huge support for my proposal, and wrote a wonderful LoR. We also got very well along on a personal level and would chat about things loosely related to academia. He is someone with a very casual style but with clear respect for boundaries. I admire him enormously, both as a scholar and as a person.
I was accepted under his supervision by a very laudatory committee. Academia being a very hierarchical world, I feel compelled to show "extra respect" to him, partly because he is famous and partly because my unconventional background sets me at a disadvantage compared to others. It seems likely that high-status academics would tacitly expect their students (i.e. without explicitly asking for it) to go to greater length for them, whether in terms of work output or in overall signs of deference. I know that they are "just human", but they are also around the top of a very special system.
I tried mentioning this to my now-supervisor, but he keeps blushing when I remind him of his academic status. He seemed to think that I was worried about my application "not being good enough" rather than about failing to show adequate deference to him. The obvious thing I could do for him would be to submit a paper before the start of my PhD. I have never written a paper before, as it seems uncommon in my field (but not unheard of) to do one that early. But he said a few times that he does not need me to do that right now, and I do not know if he means "don’t do that" or "I politely say no but would be happy if you did it anyway, and slightly disappointed if you didn’t".
I have a half-done paper addressing a recent debate around my supervisor’s main theoretical contribution, and I am confident that it can be a substantial paper if well-executed. Yet I am exhausted, and my supervisor is unlikely to help me on it before I start my PhD. What should I do?
EDIT: To clarify, this is a question about the "extra" that famous professors may expect from their PhD students. In my case, I went for a writing a paper on a subject of my choice, then submitting it to a peer-reviewed journal and hopefully avoiding desk-rejection. It feels like a huge task, but how to do it is not really the issue at hand. I would happily have presented at a major conference instead, but the conference I was accepted to did not manage to move online. A couple editors from reputable journals encouraged me to submit a paper as a master’s student, so I assumed it was a tough but realistic thing I could as that little "extra" for my supervisor.
For context, I studied in France and my supervisor is in/from Northern Europe. My field is in the social sciences.