We present a detailed analysis of the specific star formation rate -- stellar mass ($mathrm{sSFR}-M_*$) of $zle 0.13$ disk central galaxies using a morphologically selected mass-complete sample ($M_* ge 10^{9.5} M_{odot}$). Considering samples of grouped and ungrouped galaxies, we find the $mathrm{sSFR}-M_*$ relations of disk-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 $gtrsim5$. We further detect a gradual evolution of the $mathrm{sSFR}-M_*$ relation of non-grouped (field) central disk galaxies with redshift, even over a $Delta z approx 0.04$ ($approx5cdot10^{8}mathrm{yr}$) 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 disks, with the inflow being determined by the product of the cosmological accretion rate and a fuelling-efficiency -- $dot{M}_{mathrm{b,halo}}zeta$. In particular, maintaining the paradigm requires $dot{M}_{mathrm{b,halo}}zeta$ to be independent of the mass $M_{mathrm{halo}}$ of the host DMH. Furthermore, it requires the fuelling-efficiency $zeta$ to have a strong redshift dependence ($propto (1+z)^{2.7}$ for $M_*=10^{10.3} M_{odot}$ over $z=0 - 0.13$), even though no morphological transformation to spheroids can be invoked to explain this in our disk-dominated sample. The physical mechanisms capable of giving rise to such dependencies of $zeta$ on $M_{mathrm{halo}}$ and $z$ for disks are unclear.
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