  Starburst galaxies, which are known as "reservoirs" of high-energy
cosmic-rays, can represent an important high-energy neutrino "factory"
contributing to the diffuse neutrino flux observed by IceCube. In this paper,
we revisit the constraints affecting the neutrino and gamma-ray hadronuclear
emissions from this class of astrophysical objects. In particular, we go beyond
the standard prototype-based approach leading to a simple power-law neutrino
flux, and investigate a more realistic model based on a data-driven blending of
spectral indexes, thereby capturing the observed changes in the properties of
individual emitters. We then perform a multi-messenger analysis considering the
extragalactic gamma-ray background (EGB) measured by Fermi-LAT and different
IceCube data samples: the 7.5-year High-Energy Starting Events (HESE) and the
6-year high-energy cascade data. Along with starburst galaxies, we take into
account the contributions from blazars and radio galaxies as well as the
secondary gamma-rays from electromagnetic cascades. Remarkably, we find that,
differently from the highly-constrained prototype scenario, the spectral index
blending allows starburst galaxies to account for up to $40\%$ of the HESE
events at $95.4\%$ CL, while satisfying the limit on the non-blazar EGB
component. Moreover, values of $\mathcal{O}(100~\mathrm{PeV})$ for the maximal
energy of accelerated cosmic-rays by supernovae remnants inside the starburst
are disfavoured in our scenario. In broad terms, our analysis points out that a
better modeling of astrophysical sources could alleviate the tension between
neutrino and gamma-ray data interpretation.
