The pseudo-arc is the unique hereditarily indecomposable chainable continuum.
The Lelek fan is the unique compact, connected subset of the Cantor fan (the cone over the Cantor set) with a dense endpoint set. The Lelek fan is a decomposable continuum and thus does not embed into the pseudo-arc.
Question. Does the set $E(L)$ of endpoints of the Lelek fan embed into the pseudo-arc?
This is equivalent to asking whether the pseudo-arc contains Erdos space $\mathfrak E$, as each of $E(L)$ and $\mathfrak E$ contains the other.