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?

This would produce a counterexample to Problem 76 in the Houston Problem Book, which is still open I think.

The Continuum Hypothesis may be necessary.