Suppose that $X$ is a compact convex subset of a topological vector space. Suppose also that the extremal points of $X$ have the additional property that any sequence $x_n$ of extremal points has a subsequence $x_{n_k}$ converging to an extremal point $x$. Does this imply that $X$ is sequentially compact? If not, what additional conditions would imply $X$ is sequentially compact?

If the topological vector space is first countable, then the compactness of $X$ implies that it is sequentially compact as well, so we might as well assume that $X$ is not first countable. This question is a little out of my area of expertise, so I'm not sure if people ever deal with convex sets in vector spaces which are not first countable.