$\newcommand\ext{\operatorname{ext}}\newcommand\R{\mathbb R}$Let $A$ be a convex polytope in $\R^n$. Consider the closed convex cone
$$K_A:=\{(l,t)\in(\R^n)'\times\R\colon\, l(x)\ge t\ \forall x\in A\}.$$

Claim: If $(l,t)$ is on an extreme ray of $K_A$, then the hyperplane $l^{-1}(\{0\})$ is parallel to a face of the polytope $A$.

Is this claim true?