  In this note, we prove that finite CAT(0) cube complexes can be reconstructed
from their boundary distances (computed in their 1-skeleta). This result was
conjectured by Haslegrave, Scott, Tamitegama, and Tan (2023). The
reconstruction of a finite cell complex from the boundary distances is the
discrete version of the boundary rigidity problem, which is a classical problem
from Riemannian geometry. In the proofs, we use the bijection between CAT(0)
cube complexes and median graphs and the corner peelings of median graphs.
