# Banach space with dual not a GT space

Let $$X$$ be a Banach space. A bounded linear map $$u:X\to\ell_2$$ is said to be $$1$$-summing if for all finite sequence $$(x_i)\subseteq X$$ there is a constant $$C>0$$ such that $$\sum\|ux_i\|\leq C\sup\Big\{\sum|x^*(x_i)|_2:\|x^*\|_{X^*}\leq 1\Big\}.$$ A Banach space is said to be satisfy Grothendieck's theorem (in short G. T. space) if any operator $$u:X\to\ell_2$$ is $$1$$-summing. My question is the following. Does there exist a G. T. space space whose dual is not a G. T. space?

The form of Grothendieck's theorem that gives rise to the terminology "GT-space" is the fact that every bounded operator from $$\ell_1$$ to $$\ell_2$$ is 1-summing.
is every bounded operator $$\ell_\infty \to\ell_2$$ 1-summing?
Now recall that every separable Banach space embeds isomorphically into $$\ell_\infty$$ by a Hahn-Banach argument (I think this embedding can be made isometric, but we dont need that). Also recall that the p-summing norm defines an operator ideal: if $$S:X\to Y$$ is p-summing then so is $$RST:W\to Z$$ for any bounded operators $$T:W\to X$$ and $$R:Y\to Z$$.
Consequently: if every bounded operator $$\ell_\infty\to \ell_2$$ is p-summing, for some $$1\leq p<\infty$$, then every $$X\to \ell_2$$ is p-summing for any choice of separable Banach space $$X$$. In particular the identity operator on $$\ell_2$$ would be p-summing. But it is known that on any infinite-dimensional Banach space the identity operator cannot be p-summing for any $$1\leq p<\infty$$.