Co-authoring dies in "we should sync." You already know the table of contents. Somebody has to take the middle.
Assign ownership
Chapters 1–3 you, 4–6 them, 7 together. What "done" means — draft they can attack, or polished. The date. Handing off is the same shape when the other person must survive you; here they must survive the month.
Ask for a decision, not a riff
"Does this split work?" is yes or no, with a counter-split if no. It is not an invitation to reopen the thesis. If they want to reopen it, they will. Do not do that work for them in the first letter.
Keep the session off the page
The chat where you hated chapter 2's title is not their problem. The briefing is the split. Your note is the one sentence you would stand behind if it were read aloud — because it will be, by them, later, when you both forgot who agreed.