The Hubble constant ($H_0$) estimated from the local Cepheid-supernova (SN)
distance ladder is in 3-$\sigma$ tension with the value extrapolated from
cosmic microwave background (CMB) data assuming the standard cosmological
model. Whether this tension represents new physics or systematic effects is the
subject of intense debate. Here, we investigate how new, independent $H_0$
estimates can arbitrate this tension, assessing whether the measurements are
consistent with being derived from the same model using the posterior
predictive distribution (PPD). We show that, with existing data, the inverse
distance ladder formed from BOSS baryon acoustic oscillation measurements and
the Pantheon SN sample yields an $H_0$ posterior near-identical to the Planck
CMB measurement. The observed local distance ladder value is a very unlikely
draw from the resulting PPD. Turning to the future, we find that a sample of
$\sim50$ binary neutron star "standard sirens" (detectable within the next
decade) will be able to adjudicate between the local and CMB estimates.