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Is the sample mean a better point estimate of the population median than the sample median?

A beginner's question to check I've understood correctly. A basic stats textbook says:

"The variance of the sampling distribution of the median is greater than that of the sampling distribution of the mean. It follows that sample mean is likely to be closer to the population mean than the sample median. Therefore, the sample mean is a better point estimate of the population mean than the sample median."

Does it follow that for distributions where the median=mean, the sample mean is a better point estimate of the population median than the sample median?