Houston Problem Book Problem 195

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Problem 195 (HPB): Let $X_1,X_2,\ldots$ be an inverse sequence of polyhedra with bonding maps $P_n \colon X_{n+1} \rightarrow X_n$ such that the inverse limit is a hereditarily indecomposable continuum. Let $F_n$ be a continuous mapping of the $2$-sphere into $X_n$ such that $F_n$ is homotopic to the composite $P_n[F_{n+1}]$ for $n=1,2,\ldots$. Is $F_1$ homotopic to a constant mapping? (Asked by J. Krasinkiewicz 12 June 1986)

Answer: Unknown