Recall the Bishop-Phelps Theorem.

Bishop-Phelps Theorem:Let $B\subseteq E$ be a bounded, closed, convex subset of a real Banach space $E.$ Then the set $$\{e^*\in E^*: e^* \text{ attains its supremum on } B \}$$ is norm-dense in the dual $E^*.$

Does the theorem hold for extreme points (slightly different version)? More precisely,

Question:Let $B\subseteq E$ be a bounded, closed and convex subset of a real Banach space $E.$ Is it true that the set $$\{e^*\in\text{ext} \left( B_{E^*} \right): e^*(e) = \|e\|\text{ for some }e\in B\}$$ norm-dense in $\text{ext}\left( B_{E^*} \right)?$

If there is an affirmative answer to my question, may I have reference?

*Notation:* $E^*$ is the conjugate space. $B_{E^*}$ is the closed unit ball of $E^*$. We say that $e^*$ is an extreme point of $B_{E^*}$ if it cannot be expressed as midpoint of two elements from $B_{E^*}.$
Denote $\text{ext}\left( B_{E^*}\right)$ to be the set of all extreme points of $B_{E^*}.$