The presence of stable, compact circumbinary discs of gas and dust around post-asymptotic giant branch (post-AGB) binary systems has been well established. We focus on one such system: IRAS 08544-4431. We present an interferometric multi-wavelength analysis of the circumstellar environment of IRAS 08544-4431. The aim is to constrain different contributions to the total flux in the H, K, L, and N-bands in the radial direction. The data from VLTI/PIONIER, VLTI/GRAVITY, and VLTI/MATISSE range from the near-infrared, where the post-AGB star dominates, to the mid-infrared, where the disc dominates. We fitted two geometric models to the visibility data to reproduce the circumbinary disc: a ring with a Gaussian width and a flat disc model with a temperature gradient. The flux contributions from the disc, the primary star (modelled as a point-source), and an over-resolved component are recovered along with the radial size of the emission, the temperature of the disc as a function of radius, and the spectral dependencies of the different components. The trends of all visibility data were well reproduced with the geometric models. The near-infrared data were best fitted with a Gaussian ring model while the mid-infrared data favoured a temperature gradient model. This implies that a vertical structure is present at the disc inner rim, which we attribute to a rounded puffed-up inner rim. The N-to-K size ratio is 2.8, referring to a continuous flat source, analogues to young stellar objects. By combining optical interferometric instruments operating at different wavelengths we can resolve the complex structure of circumstellar discs and study the wavelength-dependent opacity profile. A detailed radial, vertical, and azimuthal structural analysis awaits a radiative transfer treatment in 3D to capture all non-radial complexity.
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