  The muon capture reaction $\mu^- + d \rightarrow n + n + \nu_{\mu}$ in the
doublet hyperfine state is studied using nuclear potentials and consistent
currents derived in chiral effective field theory, which are local and
expressed in coordinate-space (the so-called Norfolk models). Only the largest
contribution due to the $^1S_0$ $nn$ scattering state is considered. Particular
attention is given to the estimate of the theoretical uncertainty, for which
four sources have been identified: (i) the model dependence, (ii) the chiral
order convergence for the weak nuclear current, (iii) the uncertainty in the
single-nucleon axial form factor, and (iv) the numerical technique adopted to
solve the bound and scattering $A=2$ systems. This last source of uncertainty
has turned out essentially negligible. The $^1S_0$ doublet muon capture rate
$\Gamma^D(^1S_0)$ has been found to be $\Gamma^D(^1S_0)=255.8(0.6)(4.4)(2.9)$
s$^{-1}$, where the three errors come from the first three sources of
uncertainty. The value for $\Gamma^D(^1S_0)$ obtained within this local chiral
framework is compared with previous calculations and found in very good
agreement.
