Is the sum $C+\mathbb{R}^n_-$ of a closed cone $C$ and the nonpositive orthant $\mathbb{R}^n_-$ closed?

Question

A subset $C$ of $\mathbb{R}^n$ is a cone if it is closed under linear combinations with nonnegative coefficients, that is $x,y \in C$, $\lambda,\mu \geqq 0 \Rightarrow \lambda x+ \mu y\in C$. Clearly, a cone is a convex set. A cone is a closed cone if it is a closed set. In particular, the nonpositive orthant $\mathbb{R}^n_- = \{x\in\mathbb{R}^n \,\vert\, x\leqq 0\}$ is a closed cone (here $x\leqq 0$ means that $x_i\leqq 0$ for all $i=1,\dots, n$). For cones $C, D$ we define the sum $C+D =\{x+y\,\vert\, x\in C, y\in D \}$. Clearly, $C+D$ is again a cone. It is known that $C+D$ need not be closed, even if both $C$ and $D$ are closed (there are counterexamples), but perhaps it is true that if $C$ is a closed cone, and $D=\mathbb{R}^n_-$, then the sum $C+D$ is closed. I would be grateful for either a proof or a counterexample.