We study the evolution of kinematically defined stellar discs in 10 Fornax-like clusters identified in the TNG50 run from the IllustrisTNG suite of cosmological simulations. We considered disc galaxies with present-day stellar mass M⋆ ≥ 3 × 108 M⊙ and follow their evolution since first entering their host cluster. Very few stellar discs survive since falling in such dense environments, ranging from 40 per cent surviving to all being disrupted. Such survival rates are consistent with what reported earlier for the two more massive, Virgo-like clusters in TNG50. In absolute terms, however, the low number of present-day disc galaxies in Fornax-like clusters could be at odds with the presence of three edge-on disc galaxies in the central regions of the actual Fornax cluster, as delineated by the Fornax3D survey. When looking at the Fornax analogues from random directions and with the same selection function of Fornax3D, the probability of finding three edge-on disc galaxies in any one Fornax-like cluster in TNG50 is rather low, albeit not impossible. We also compared the stellar-population properties near the equatorial plane derived from integral-field spectroscopy for the three edge-ons in Fornax to similar line-of-sight integrated values for present-day disc galaxies in TNG50. For one of these, the very old and metal-rich stellar population of its disc cannot be matched by any the disc galaxies in TNG50, including objects in the field. We discuss possible interpretations of these findings, while pointing to future studies on passive cluster spirals as a way to further test state-of-the-art cosmological simulations.