Let be the angle through which the people turn with respect to the laboratory.
Let be the angle through which the disk turns with respect to the laboratory.
because speed is constant
angular momentum of people
angular momentum of disk = - angular momentum of people
Now at the completion of one revolution across the surface of the disk
You now have two equations in two variables phi_d and phi_p so you can solve them. Though I get a different answer to your spoiler, perhaps I made a silly algebraic mistake above.