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We extend our studies of a new class of $2 \to 3$ exclusive processes usingthe collinear factorisation framework by considering the exclusivephotoproduction of a $\gamma\,\rho$ pair, in the kinematics where the pair hasa large invariant mass, {and the outgoing $ \rho $-meson has a sufficientlylarge transverse momentum to not resonate with the nucleon}. We cover the wholekinematical range from medium energies in fixed target experiments to verylarge energies of colliders, by considering the experimental conditions ofJLab~12-GeV, COMPASS, future EIC and LHC (in ultra-peripheral collisions)cases. One of the main interest in studying the present process is that itprovides access to both chiral-even and chiral-odd GPDs, depending on thepolarisation of the outgoing $\rho$-meson, both at leading twist. Our analysiscovers both neutral and charged $ \rho $-mesons. We find that the order ofmagnitude of the obtained cross sections are sufficiently large for a dedicatedexperimental analysis to be performed, especially at JLab, for bothlongitudinally and transversely polarised $\rho$. Furthermore, we compute thelinear photon beam polarisation asymmetry {which is} sizeable for alongitudinally polarised meson. These predictions are obtained for bothasymptotic distribution amplitude (DA) and the \textit{holographic} DA.

Probing chiral-even and chiral-odd leading twist quark generalised  parton distributions through the exclusive photoproduction of a $ \gamma \rho  $ pair