  The KBC void is a local underdensity with the observed relative density
contrast $\delta \equiv 1 - \rho/\rho_{0} = 0.46 \pm 0.06$ between 40 and 300
Mpc around the Local Group. If mass is conserved in the Universe, such a void
could explain the $5.3\sigma$ Hubble tension. However, the MXXL simulation
shows that the KBC void causes $6.04\sigma$ tension with standard cosmology
($\Lambda$CDM). Combined with the Hubble tension, $\Lambda$CDM is ruled out at
$7.09\sigma$ confidence. Consequently, the density and velocity distribution on
Gpc scales suggest a long-range modification to gravity. In this context, we
consider a cosmological MOND model supplemented with $11 \, \rm{eV}/c^{2}$
sterile neutrinos. We explain why this $\nu$HDM model has a nearly standard
expansion history, primordial abundances of light elements, and cosmic
microwave background (CMB) anisotropies. In MOND, structure growth is
self-regulated by external fields from surrounding structures. We constrain our
model parameters with the KBC void density profile, the local Hubble and
deceleration parameters derived jointly from supernovae at redshifts $0.023 -
0.15$, time delays in strong lensing systems, and the Local Group velocity
relative to the CMB. Our best-fitting model simultaneously explains these
observables at the $1.14\%$ confidence level (${2.53 \sigma}$ tension) if the
void is embedded in a time-independent external field of ${0.055 \, a_{_0}}$.
Thus, we show for the first time that the KBC void can naturally resolve the
Hubble tension in Milgromian dynamics. Given the many successful a priori MOND
predictions on galaxy scales that are difficult to reconcile with $\Lambda$CDM,
Milgromian dynamics supplemented by $11 \, \rm{eV}/c^{2}$ sterile neutrinos may
provide a more holistic explanation for astronomical observations across all
scales.
