  We study the behaviour of the streamwise velocity variance in turbulent
wall-bounded flows using a DNS database of pipe flow up to $Re_{\tau} \approx
12000$. The analysis of the spanwise spectra in the viscous near-wall region
strongly hints to the presence of an overlap layer between the inner- and the
outer-scaled spectral ranges, featuring a $k_{\theta}^{-1+\alpha}$ decay (with
$k_{\theta}$ the wavenumber in the azimuthal direction, and $\alpha \approx
0.18$), hence shallower than suggested by the classical formulation of the
attached-eddy model. The key implication is that the contribution to the
streamwise velocity variance from the largest scales of motion
(superstructures) slowly declines as $Re_{\tau}^{-\alpha}$, and the integrated
variance follows a defect power law of the type $\left< u^2 \right>^+ = A - B
\, Re_{\tau}^{-\alpha}$, with constants $A$ and $B$ depending on $y^+$. The DNS
data very well support this behaviour, which implies that strict wall scaling
is restored in the infinite Reynolds number limit. The extrapolated limit
distribution of the streamwise velocity variance features a buffer-layer peak
value of $\left< u^2 \right>^+ \approx 12.1$, and an additional outer peak with
larger magnitude. The analysis of the velocity spectra also suggests a similar
behaviour of the dissipation rate of the streamwise velocity variance at the
wall, which is expected to attain a limiting value of about $0.28$, hence
slightly exceeding the value $0.25$ which was assumed in previous
analyses~\citep{chen_21}. We have found evidence suggesting that the reduced
near-wall influence of wall-attached eddies is likely linked to the formation
of underlying turbulent Stokes layers.
