Indicator functions of the streamwise normal-stress profiles (NSP), based on careful differentiation of some of the best direct numerical simulations (DNS) data from channel and pipe flows, over the range 550 < R e τ < 16 000 , are examined to establish the existence and range in wall distances of either a logarithmic-trend segment or a 1/4-power region. For nine out of 15 cases of DNS data we examined where R e τ < 2000 , the NSP did not contain either of the proposed trends. As R e τ exceeds around 2000 a 1/4-power, reflecting the “bounded-dissipation” predictions of Chen and Sreenivasan [“Law of bounded dissipation and its consequences in turbulent wall flows,” J. Fluid Mech. 933, A20 (2022); “Reynolds number asymptotics of wall-turbulence fluctuations,” J. Fluid Mech. 976, A21 (2023)] and data analysis of Monkewitz [“Reynolds number scaling and inner-outer overlap of stream-wise Reynoldss stress in wall turbulence,” arXiv:2307.00612 (2023)], develops near y + = 1000 and expands with Reynolds numbers extending to 1000 < y + < 10 000 for R e τ around 15 000. This range of 1/4-power NSP corresponds to a range of outer-scaled Y between around 0.3 and 0.7. The computational database examined did not include the zero-pressure-gradient boundary layer experiments at higher Reynolds numbers where the logarithmic trend in the NSP has been previously reported around y+ of 1000 by Marusic et al. [“Attached eddy model of wall turbulence,” Annu. Rev. Fluid Mech. 51, 49-74 (2019); “The logarithmic variance of streamwise velocity and conundrum in wall turbulence,” J. Fluid Mech. 933, A8 (2022)] according to a “wall-scaled eddy model.”