We present a detailed analysis of the specific star formation rate-stellar mass (sSFR-M*) of z ≤ 0.13 disc central galaxies using a morphologically selected mass-complete sample (M* ≥ 109.5M⊙). Considering samples of grouped and ungrouped galaxies, we find the sSFR-M* relations of disc-dominated central galaxies to have no detectable dependence on host dark-matter halo (DMH) mass, even where weak-lensing measurements indicate a difference in halo mass of a factor ≳ 5.We further detect a gradual evolution of the sSFR-M* relation of non-grouped (field) central disc galaxies with redshift, even over a Δz ≈ 0.04 (≈5 × 108 yr) interval, while the scatter remains constant. This evolution is consistent with extrapolation of the 'main sequence of star-forming-galaxies' from previous literature that uses larger redshift baselines and coarser sampling. Taken together, our results present new constraints on the paradigm under which the SFR of galaxies is determined by a self-regulated balance between gas inflows and outflows, and consumption of gas by star formation in discs, with the inflow being determined by the product of the cosmological accretion rate and a fuelling efficiency - -Mb,haloζ. In particular, maintaining the paradigm requires -Mb,haloζ to be independent of themass Mhalo of the host DMH. Furthermore, it requires the fuelling efficiency ζ to have a strong redshift dependence (∝(1 + z)2.7 forM* = 1010.3M⊙ over z = 0-0.13), even though no morphological transformation to spheroids can be invoked to explain this in our disc-dominated sample. The physical mechanisms capable of giving rise to such dependencies of ζ on Mhalo and z for discs are unclear.