  We provide the eigenfunctions for a quantum chain of $N$ conformal spins with
nearest-neighbor interaction and open boundary conditions in the irreducible
representation of $SO(1,5)$ of scaling dimension $\Delta = 2 - i \lambda$ and
spin numbers $\ell=\dot{\ell}=0$. The spectrum of the model is separated into
$N$ equal contributions, each dependent on a quantum number $Y_a=[\nu_a,n_a]$
which labels a representation of the principal series. The eigenfunctions are
orthogonal and we computed the spectral measure by means of a new star-triangle
identity. Any portion of a conformal Feynmann diagram with square lattice
topology can be represented in terms of separated variables, and we reproduce
the all-loop "fishnet" integrals computed by B. Basso and L. Dixon via
bootstrap techniques. We conjecture that the proposed eigenfunctions form a
complete set and provide a tool for the direct computation of conformal data in
the fishnet limit of the supersymmetric $\mathcal{N}=4\,$ Yang-Mills theory at
finite order in the coupling, by means of a cutting-and-gluing procedure on the
square lattice.
