Is there a dense connected subset $X$ of the Cantor fan $$(C\times [0,1])/(C\times \{1\})$$ such that for every two connected subsets $X_1,X_2\subseteq X$, the intersection $X_1\cap X_2$ is connected?

If so, $X$ could be used to construct a counterexample to Problem 76 in the [Houston Problem Book][1], which is still open I think.


  [1]: http://hyperspacewiki.org/index.php/The_Houston_Problem_Book_Problems_51-100