  We show that the anomalous dimension $\gamma_G$ of the scalar glueball
operator contains information on the mechanism that leads to the onset of
conformality at the lower edge of the conformal window in a non-Abelian gauge
theory. In particular, it distinguishes whether the merging of an UV and an IR
fixed point -- the simplest mechanism associated to a conformal phase
transition and preconformal scaling -- does or does not occur. At the same
time, we shed light on new analogies between QCD and its supersymmetric
version. In SQCD, we derive an exact relation between $\gamma_G$ and the mass
anomalous dimension $\gamma_m$, and we prove that the SQCD exact beta function
is incompatible with merging as a consequence of the $a$-theorem; we also
derive the general conditions that the latter imposes on the existence of fixed
points, and prove the absence of an UV fixed point at nonzero coupling above
the conformal window of SQCD. Perhaps not surprisingly, we then show that an
exact relation between $\gamma_G$ and $\gamma_m$, fully analogous to SQCD,
holds for the massless Veneziano limit of large-N QCD. We argue, based on the
latter relation, the $a$-theorem, perturbation theory and physical arguments,
that the incompatibility with merging may extend to QCD.
