  Context. Rocky planets form by the concentration of solid particles in the
inner few au regions of planet-forming disks. Their chemical composition
reflects the materials in the disk available in the solid phase at the time the
planets were forming. Aims. We aim to constrain the structure and dust
composition of the inner disk of the young star HD 144432, using an extensive
set of infrared interferometric data taken by the Very Large Telescope
Interferometer (VLTI), combining PIONIER, GRAVITY, and MATISSE observations.
Methods. We introduced a new physical disk model, TGMdust, to image the
interferometric data, and to fit the disk structure and dust composition. We
also performed equilibrium condensation calculations with GGchem. Results. Our
best-fit model has three disk zones with ring-like structures at 0.15, 1.3, and
4.1 au. Assuming that the dark regions in the disk at ~0.9 au and at ~3 au are
gaps opened by planets, we estimate the masses of the putative gap-opening
planets to be around a Jupiter mass. We find evidence for an optically thin
emission ($\tau<0.4$) from the inner two disk zones ($r<4$ au) at $\lambda>3\
\mu$m. Our silicate compositional fits confirm radial mineralogy gradients. To
identify the dust component responsible for the infrared continuum emission, we
explore two cases for the dust composition, one with a silicate+iron mixture
and the other with a silicate+carbon one. We find that the iron-rich model
provides a better fit to the spectral energy distribution. Conclusions. We
propose that in the warm inner regions ($r<5$ au) of typical planet-forming
disks, most if not all carbon is in the gas phase, while iron and iron sulfide
grains are major constituents of the solid mixture along with forsterite and
enstatite. Our analysis demonstrates the need for detailed studies of the dust
in inner disks with new mid-infrared instruments such as MATISSE and JWST/MIRI.
