On the extreme points of two convex sets

Question

Let $A$ and $B$ be two compact convex sets (which may be assumed to be polytopes) in $\mathbb R^n$ such that $A\cap B\ne\emptyset$. Is it then always true that either $A\cap\text{ext}B\ne\emptyset$ or $B\cap\text{ext}A\ne\emptyset$, where $\text{ext}$ denotes the set of all extreme points of a set?

Answer 1

No: consider the line segments $\{0\}\times[-1,1]$ and $[-1,1]\times\{0\}$ in $\mathbb{R}^2$.