Is "$A \subset X^*$ separates points of $X$ iff. $\overline{\operatorname{conv}}^{w^*}(A) = X^*$" a correct statement?

Question

Is "$A \subset X^*$ separates points of $X$ iff. $\overline{\operatorname{conv}}^{w^*}(A) = X^*$" a correct statement?

I saw this statement in a question in a book but it didn't seem right to me, so I just want to make sure that I'm not missing anything.

Counterexample: $B_{X^*}$ separates points of $X$ since if $x \neq 0$ then $||x|| = \sup_{x^* \in B_{X^*}} x^*(x) > 0$, but $\overline{\operatorname{conv}}^{w^*}(B_{X^*}) = B_{X^*}$ since $B_{X^*}$ is convex and $w^*$-compact.

Thanks.