Binary post-asymptotic giant branch (post-AGB) stars have orbital periods in the range of 100--2500 days in eccentric orbits. They are surrounded by circumbinary dusty discs. They are the immediate result of unconstrained binary interaction processes. Their observed orbital properties do not correspond to model predictions: Neither the periods nor the high eccentricities are expected. Our goal is to investigate if interactions between a binary and its circumbinary disc during the post-AGB phase can result in their eccentric orbits, while simultaneously explaining the chemical anomaly known as depletion. For this paper, we selected three binaries (EP Lyr, RU Cen, HD 46703) with well-constrained orbits, luminosities, and chemical abundances. We used the MESA code to evolve post-AGB models, while including the accretion of metal-poor gas. This allows us to constrain the evolution of the stars and study the impact of circumbinary discs on the orbital properties of the models. We investigate the effect of torques produced by gas inside the binary cavity and the effect of Lindblad resonances on the orbit, while also including the tidal interaction following the equilibrium tide model. We find that none of our models are able to explain the high orbital eccentricities of the binaries in our sample. The accretion torque does not significantly impact the binary orbit, while Lindblad resonances can pump the eccentricity up to only $e approx 0.2$. At higher eccentricities, the tidal interaction becomes too strong, so the high observed eccentricities cannot be reproduced. However, even if we assume tides to be ineffective, the eccentricities in our models do not exceed $approx 0.25$. We conclude that either our knowledge of disc-binary interactions is still incomplete, or the binaries must have left their phase of strong interaction in an eccentric orbit.
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