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 $Y_a=[ u_a,n_a]$ 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.
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