  We discuss the uncertainties in constraining low-energy constants of chiral
effective field theory from $^3$H $\beta$ decay. The half-life is very
precisely known, so that the Gamow-Teller matrix element has been used to fit
the coupling $c_D$ of the axial-vector current to a short-range two-nucleon
pair. Because the same coupling also describes the leading one-pion-exchange
three-nucleon force, this in principle provides a very constraining fit,
uncorrelated with the $^3$H binding energy fit used to constrain another
low-energy coupling in three-nucleon forces. However, so far such $^3$H
half-life fits have only been performed at a fixed cutoff value. We show that
the cutoff dependence due to the regulators in the axial-vector two-body
current can significantly affect the Gamow-Teller matrix elements and
consequently also the extracted values for the $c_D$ coupling constant. The
degree of the cutoff dependence is correlated with the softness of the employed
NN interaction. As a result, present three-nucleon forces based on a fit to
$^3$H $\beta$ decay underestimate the uncertainty in $c_D$. We explore a range
of $c_D$ values that is compatible within cutoff variation with the
experimental $^3$H half-life and estimate the resulting uncertainties for
many-body systems by performing calculations of symmetric nuclear matter.
