  The spectral shape of forbidden beta-decays is a crucial benchmark for
nuclear physics calculations and has important implications also for
astroparticle physics experiments. Among the interesting isotopes in this
field, $^{115}$In is an excellent candidate, being a primordial nuclide
featuring a good compromise among Q-value (497.954 keV) and half-life
(4.41$\times$10$^{14}$ yr). In this paper, we propose to exploit a cryogenic
calorimeter based on In$_2$O$_3$ crystal to perform a high-precision
measurement of the $\beta$-decay energy spectrum, discussing also the results
obtained within a preliminary test of this crystal and the next steps to
improve the detector performance.
