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.
Note that $X$ minus the vertex point must be totally disconnected, having only one point in each blade of the fan. I constructed a connected set with this property by taking the complete Erdös space in the Cantor fan, and moving its points up and down the blades using a sin function to make it dense.
The Continuum Hypothesis may be necessary to construct an example. For if $X$ is an example then for every two dense connected subsets $X_1,X_2\subseteq X$ we must have $|X_1\cap X_2|>1$. Essentially the only example I know like this comes from Miller's biconnected set, which needs the Continuum Hypothesis.