I'd think that should be enough to get quite a few layers, sure maybe some minimal space between layers for cooling, but radically less than the non-superconducting designs.
[1] https://techcrunch.com/2019/08/19/the-five-technical-challen...
Edit: I assume a supercoducting microprocessor would use a strategy similar to the AI monolith in [1]. Just fuse off and route around errors on a contiguous wafer and distribute the computation to exploit the dark zones for heat dissipation.
The thickness of the required insulating barrier presents a hard lower limit to the structure size.
The actual value of that limit depends on the material used and the particular implementation of the Josephson junction, of which there seems to be quite a few.
So the limit depends on how thin the barrier can be made.