  In a previous article, we have shown that the discrepancy between the
fixed-order (FOPT) and contour-improved (CIPT) perturbative expansions for
$\tau$ hadronic spectral function moments, which had affected the precision of
$\alpha_s$ determinations for many years, may be reconciled by employing a
renormalon-free (RF) scheme for the gluon condensate (GC) matrix element. In
addition, the perturbative convergence of spectral function moments with a
sizeable GC correction can be improved. The RF GC scheme depends on an IR
factorization scale $R$ and the normalization $N_g$ of the GC renormalon. In
the present work, we use three different methods to determine $N_g$, yielding a
result with an uncertainty of $40\%$. Following two recent state-of-the-art
strong coupling determination analyses at ${\cal O}(\alpha_s^5)$, we show that
using the renormalon-free GC scheme successfully reconciles the results for
$\alpha_s({m_\tau^2})$ based on CIPT and FOPT. The uncertainties due to
variations of $R$ and the uncertainty of $N_g$ only lead to a small or moderate
increase of the final uncertainty of $\alpha_s(m_\tau^2)$, and affect mainly
the CIPT expansion method. The FOPT and CIPT results obtained in the RF GC
scheme may be consistently averaged. The RF GC scheme thus constitutes a
powerful new ingredient for future analyses of $\tau$ hadronic spectral
function moments.
