In this paper we consider systems of quantum particles in the $4d$ Euclidean space which enjoy conformal symmetry. The algebraic relations for conformal-invariant combinations of positions and momenta are used to construct a solution of the Yang-Baxter equation in the unitary irreducibile representations of the principal series $Delta=2+i u$ for any left/right spins $ell,dot{ell}$ of the particles. Such relations are interpreted in the language of Feynman diagrams as integral emph{star-triangle} identites between propagators of a conformal field theory. We prove the quantum integrability of a spin chain whose $k$-th site hosts a particle in the representation $(Delta_k,ell_k, dot{ ell}_k)$ of the conformal group, realizing a spinning and inhomogeneous version of the quantum magnet used to describe the spectrum of the bi-scalar Fishnet theories. For the special choice of particles in the scalar $(1,0,0)$ and fermionic $(3/2,1,0)$ representation the transfer matrices of the model are Bethe-Salpeter kernels for the double-scaling limit of specific two-point correlators in the $gamma$-deformed $mathcal{N}=4$ and $mathcal{N}=2$ supersymmetric theories.
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