The thread "Are inconsistent estimators ever preferable?" and @whuber's answer in it shows that there exists an inconsistent estimator that can outperform a reasonable consistent one for all finite $n$, for a suitable loss function. @whuber's idea for constructing an example of interest is based on finding a loss function that is minimized not at the true parameter value but elsewhere.

I would like to take the problem one step further. In examples similar to @whuber's, there may exist a pseudo-true (for lack of a better term) parameter value that minimizes the loss function in population. (This value depends on the loss function and the actual parameter value, I think.)

Question: If we have a reasonable consistent estimator for the pseudo-true value (corresponding to a given loss function and the actual parameter value), are there examples of an inconsistent estimator which outperforms it for all finite $n$ with respect to the same loss function?