Houston Problem Book Problem 2

Problem 2 (HPB): Suppose $X$ is a continuum such that for each positive number $\epsilon$ there are at most finitely many pairwise disjoint connected sets in $X$ of diameter greater than $\epsilon$. Suppose, if $Y$ is any continuous, monotonic, Hausdorff image of $X$, then $Y$ can be embedded in a continuum $Z$ which is the union of a countable family of arcs. Is every connected subset of $X$ path connected?