We have an analytic solution in the supplementary materials [1]. One of the biggest challenges to optimality proofs is how to capture the importance of geography as a constraint. I don't believe there are any papers on redistricting that include geographic constraints in analytic/formal solutions. That's why we prefer simulations for our main results.
[1]: https://static.cambridge.org/content/id/urn%3Acambridge.org%...