Suppose I have a closed convex cone $C\subseteq \mathbb R^n$ and suppose that for every $x$ in the non-negative orthant $\mathbb R_{0+}^n$ there is a $y\in C$ such that $x\cdot y>0$ (with the standard scalar product). Does it follow that intersection of $C$ with the non-negative orthant contains more than just the origin?

This is true for $n=2$.