  We reexamine the nature of the metallic phase of the one-dimensional
half-filled Holstein model of spinless fermions. To this end we determine the
Tomonaga-Luttinger-liquid correlation parameter $K_\rho$ by large-scale
density-matrix renormalisation-group (DMRG) calculations, exploiting (i) the
leading-order scaling relations between the ground-state energy and the
single-particle excitation gap and (ii) the static charge structure factor in
the long-wavelength limit. While both approaches give almost identical results
for intermediate-to-large phonon frequencies, we find contrasting behaviour in
the adiabatic regime: (i) $K_\rho>1$ (attractive) versus (ii) $K_\rho<1$
(repulsive). The latter result for the correlation exponent is corroborated by
data obtained for the momentum distribution function $n(k)$, which puts the
existence of an attractive metallic state in the spinless fermion Holstein
model into question. We conclude that the scaling relation must be modified in
the presence of electron-phonon interactions with noticeable retardation.
