  The scaling of Reynolds stresses in turbulent wall-bounded flows is the
subject of a long running debate. In the near-wall ``inner'' region, a sizeable
group, inspired by the ``attached eddy model'', has advocated the unlimited
growth of $\langle uu\rangle^+$ and in particular of its inner peak at
$y^+\approxeq 15$, with $\ln\Reytau$ \citep[see e.g.][and references
therein]{smitsetal2021}. Only recently, \citet{chen_sreeni2021,chen_sreeni2022}
have argued on the basis of bounded dissipation, that $\langle uu\rangle^+$
remains finite in the inner near-wall region for $\Reytau\rightarrow\infty$,
with finite Reynolds number corrections of order $\Reytau^{-1/4}$. In this
paper, the overlap between the two-term inner expansion $f_0(y^+) +
f_1(y^+)/\Reytau^{1/4}$ of \citet{monkewitz22} and the leading order outer
expansion for $\langle uu\rangle^+$ is shown to be of the form $C_0 +
C_1\,(y^+/\Reytau)^{1/4}$. With a new indicator function, overlaps of this form
are reliably identified in $\langle uu\rangle^+$ profiles for channels and
pipes, while the situation in boundary layers requires further clarification.
On the other hand, the standard logarithmic indicator function, evaluated for
the same data, shows no sign of a logarithmic law to connect an inner expansion
of $\langle uu\rangle^+$ growing as $\ln{\Reytau}$ to an outer expansion of
order unity.
