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 a Grothendieck space if any operator $u:X\to\ell_2$ is $1$-summing. My question is the following. Does there exist a Grothendieck space whose dual is not a Grothendieck 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?