MUSE-DARK IV. Comparing Dark Matter Densities to Star Formation Histories in 375 intermediate-z galaxies

(Abridged) Baryonic feedback is expected to transform cuspy dark-matter (DM) haloes into cores, but direct observational links between inner DM slopes and galaxy star-formation histories (SFHs) remain limited. We investigate and quantify the connection between halo inner slopes and baryonic assembly histories in a large sample of intermediate-z galaxies. We analyse 375 galaxies at 0.1<z<1.43 spanning 7<log(Mstar/Msun)<11.2 using 3D forward modelling of MUSE kinematics and disk-halo decompositions with mass-dependent (DC14-based) and generalised Navarro-Frenk-White (gNFW) halo profiles. Stellar population parameters are inferred from multi-band Spectral Energy Distribution fitting of UV-to-MIR photometry using a non-parametric SFH model. Bayesian model comparison shows that the DC14-based halo density profiles are statistically preferred over gNFW for a substantial fraction of the sample. The inferred inner slopes from gNFW fits nevertheless follow the DC14 expectation and reproduce the expected dependence on the stellar-to-halo mass ratio, despite this not being explicitly imposed. Partial correlation coefficient and random forest regression analyses consistently identify stellar mass (or, equivalently, the stellar-to-halo mass ratio) as the primary driver of the inner density slope, a result that holds robustly even when the analysis is restricted to the gNFW fits. In contrast, SFH-related quantities (e.g. formation time and 'burstiness' indicators) exhibit no independent predictive power on the DM halo inner slope once the dependence on stellar mass is accounted for. Cored systems preferentially occupy lower-mass, more bursty bins with later assembly times, whereas cuspy systems are associated with higher stellar masses, lower star formation activity, and earlier assembly epochs, in agreement with theoretical expectations of stellar-feedback-induced cores.