The correct answer is False.
The Central Limit Theorem (CLT) states that, given certain conditions, the sampling distribution of the mean of a sufficiently large number of iterates of independent random variables, each with a well-defined expected value and well-defined variance, will be approximately normally distributed, regardless of the underlying distribution.
The standard error of the mean is the standard deviation of the sampling distribution of the mean. It is estimated by the sample standard deviation divided by the square root of the sample size.
Replacing the standard error by its estimated value does not change the CLT. The CLT is a statement about the sampling distribution of the mean, not about the standard error of the mean. The standard error of the mean is just a statistic that is used to estimate the standard deviation of the sampling distribution of the mean.
In other words, the CLT states that the sampling distribution of the mean will be approximately normally distributed, regardless of the underlying distribution. This is true whether or not the standard error of the mean is used to estimate the standard deviation of the sampling distribution of the mean.