Solved – what is the sample correlation between the mean and the standard deviation under normality assumption

While the (usual) formula for sample standard deviation contains the sample mean (so you might say that it depends on the mean algebraically), that doesn't impact their independence in the statistical sense.

The population correlation of the sample mean and the sample standard deviation is zero because the random variables (sample mean, sample standard deviation) are independent.

Indeed $bar{X}$ and $X_i-bar{X}$ are independent and the independence of the sample variance (and similarly standard deviation) from $bar{X}$ follows from that.

Below is a plot of $x_1-bar{x}$ vs $bar{x}$ for samples of size 10 from a normal distribution; it's possible to show these are independent (and indeed, jointly normal).

The sample correlation between the random variables won't be exactly zero — it, too, is a random variable. If you gather $m$ samples of size $n$ and compute the mean and sd in each sample of size $n$ and calculate their correlation over the $m$ samples, that will have some deviation from 0 each time, but the population quantity it estimates is 0. Imagine choosing say 50 points in the plot above for example; the sample correlation of those 50 points won't be exactly 0.