Suppose that $\{\mu_n\}$ is a sequence of Borel probability measures on a compact metric space $X$ and suppose that $\{\mu_n\}$ converges weakly to a Borel probability measure $\mu$ on $X$. If $\mu$ is absolutely continuous with respect to $\mu^*$ (where $\mu^*$ is a Borel probability measure on $X$ different from $\mu$), does there exist a sequence $\{\mu^*_n\}$ of Borel probability measures on $X$ such that for each $n$, $\mu_n$ is absolutely continuous with respect to $\mu^*_n$ and the sequence $\{\mu^*_n\}$ converges weakly to $\mu^*$?