$2$-to-$1$ maps on hereditarily indecomposable continua

Suppose $f$ is a $2{\text {-to-}}1$ continuous map from the hereditarily indecomposable continuum $X$ onto a continuum $Y$. In order for it to be the case that each proper subcontinuum $C$ in $Y$ has as its preimage two disjoint continua each of which $f$ maps homeomorphically onto $C$, it is obviously necessary that $f$ satisfy the condition that each nondense connected subset of $Y$ has disconnected preimage. We show that this condition is also sufficient, and thus any $2{\text {-to-}}1$ continuous map from a hereditarily indecomposable continuum satisfying this condition must be confluent and have an image that is hereditarily indecomposable.