# $\ell^\infty / ces_0$ as an ordered Banach space

Let $$ces_0:=\{\xi\in\ell^\infty : \lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^{n}\xi_k=0\}$$ and $$q:\ell^\infty \to \ell^\infty/ces_0$$ be the usual quotient map. The space $$ces_0$$ is closed in $$\ell^\infty$$, but is not an order ideal with respect to the standard order in $$\ell^\infty$$. Therefore the quotient $$\ell^\infty/ces_0$$ need not necessarily be a Banach lattice. Still, one may equip it with the quotient cone $$q(\ell^\infty_+)$$ making $$\ell^\infty/ces_0$$ into an ordered Banach space.

Banach spaces ordered by closed cones tend to be a bit nicer, so a question I'm struggling with is:

Is the quotient cone $$q(\ell^\infty_+)$$ (norm) closed in $$\ell^\infty/ces_0$$? Or equivalently: is $$ces_0 + \ell^\infty_+$$ (norm) closed in $$\ell^\infty$$?