In regression, is functional confidence bound influenced by std err of each yi?

I am trying to fit {(xi,yi)} to a curve y = f(x) + ep(x). Each yi has std err SEi. I use weighted regression, wi = 1/SEi^2/Ni, where Ni is the number of measurements used to get yi.

If I amplified each SEi by a factor of 10, the functional confidence band of the fitted curve remains the same, because the relative weights are still the same. However, according to intuition, the functional confidence band should be greater now since the uncertainty SEi of each data point is greater.

Could someone explain this?

Thanks!