  We observe that the would-be running coupling on the lattice defined by means
of the gradient-flow method in order to identify the conformal window of QCD is
not renormalization-group invariant (RGI). Indeed, we show that the would-be
running coupling, $g_{wb}^2(t)\propto t^2\langle E(t)\rangle$, -- with $\langle
E(t)\rangle$ the expectation value of the Lagrangian density smeared by means
of the gradient flow -- has an anomalous dimension associated to the
multiplicative renormalization factor of $t^2\langle E(t)\rangle$. As a
consequence, at a nontrivial infrared (IR) fixed point with nonvanishing
anomalous dimension, $\gamma_*$, in the conformal window, the would-be running
coupling vanishes asymptotically as $g_{wb}^2(t)\propto t^2\langle
E(t)\rangle\sim t^{-\gamma_*/2}$ and does not scale as $g_{wb}^2(t)\propto
t^2\langle E(t)\rangle\sim g_{wb}^{*2}\neq 0$, with $g_{wb}^{*}$ the
nonvanishing would-be coupling at the nontrivial fixed point, as postulated in
the literature. The associated would-be beta function,
$\beta_{wb}({g}_{wb}^2(t))$, is not proportional to a true RGI beta function,
and it also vanishes asymptotically in the IR for nonvanishing $\gamma_*$ at
the IR fixed point. Moreover, $\beta_{wb}$ violates two-loop universality and
may develop spurious zeroes both in the confined and the conformal phase,
despite $g_{wb}^2(t)$ is asymptotic to a true RGI running coupling in a
neighborhood of the asymptotically free fixed point. Our analysis allows us to
reinterpret the contradictory lattice results based on this method,
specifically those for the $N_f$=12 theory, explain the origin of their
discrepancies and suggest a new strategy to discriminate between the confined
phase and the conformal window. In this respect, we disagree with a recent
claim that attributes the same contradictory results to staggered fermions
being in the wrong universality class.
