A Banach space $X$ is called indecomposable if there exists no infinite-dimensional subspaces $M$ and $N$ such that $X = M \bigoplus N$. If every infinite-dimensional closed subspace of $X$ is indecomposable, then $X$ is called Hereditarily indecomposable.
The well known "Separable Quotient problem" asks that does every Banach space has an infinite-dimensional separable quotient space？
My question is that: Does there exist an inseparable Banach space $X$ such that its dual space $X^*$ is Hereditarily indecomposable?
PS: It is easy to see that if such Banach space $X$ exists, then $X$ has no infinite-dimensional separable quotient space. Thus the well known "Separable Quotient problem" is sloved in the negative.