In my course notes on a regression course with regards to the detection of heteroskedasticity there's the following quote:
"Because the least-squares residuals have unequal variances even in the homoscedastic case, it is preferable to use the standardized residuals."
My intuition tells me that since the LS regression line necessarily goes through the center of the datacloud, it will be a better fit for points in the middle of the covariate space than on the tails, thus giving us larger variance on the extremes.
Despite this, this does not seem like it's necessary. And at the same time I wonder about why do we care for homoscedasticity on standardized or studentized residuals and not for the raw ones.