Weighted least squares (WLS) and robust standard errors are sometimes presented as alternative approaches for obtaining reliable standard errors of estimates of regression coefficients in the presence of heteroscedasticity. However, I notice that my software (gretl) offers robust standard errors as an option when using WLS.
A situation in which it seems this might be useful is where, in a regression of Y on X, there is a clear reason for heteroscedasticity, for example a scale effect such that larger values of Y are expected to be associated with larger variances. One might then use WLS, giving a higher weighting to observations with smaller Y (or, perhaps better, to observations with smaller E[Y | X], as inferred from an initial OLS regression). However, it might be found that the WLS residuals suggested some remaining heteroscedasticity that the weighting had not eliminated. This would suggest that the standard errors estimated by WLS might not be entirely reliable, and to address this one might opt for robust standard errors (rather than attempting to do so via some more complex weighting pattern).
Question: Assuming the number of observations is reasonably large (over 100, say), are there any pitfalls in using WLS with robust standard errors when estimating standard errors of regression coefficients?