A Banach space is called Hereditarily indecomposable if there exists no infinite-dimensional subspaces $M$ and $N$ such that $X = M \bigoplus N$.

The well known "Separable Quotient problem" asks that does every Banach space has an infinite-dimensional quotient space？

My question is: Does there exist an inseparable Banach space $X$ such that its dual space $X^*$ is Hereditarily indecomposable.

PS: It is easy to see taht if such Banach space $X$ exists, then $X$ has no infinite-dimensional quotient space. Thus the well known "Separable Quotient problem" is sloved in the negative.