Hyperspaces of nowhere topologically complete spaces

It is proved that if $X$ is a connected locally continuumwise connected coanalytic nowhere topologically complete space, then the hyperspace $2^X$ of all nonempty compact subsets of $X$ is strongly universal in the class of all coanalytic spaces. Moreover, $2^X$ is homeomorphic to $\Pi_2$ if $X$ is a Baire space, and to $Q\setminus\Pi_1$ if $X$ contains a dense absolute $G_\delta$-set $G\subset X$ such that the intersection $G\cap U$ is connected for any open connected $U\subset X$. (Here $\Pi_1,\Pi_2\subset X$ are the standard subsets of the Hilbert cube $Q$ absorbing for the classes of analytic and coanalytic spaces, respectively.) Similar results are obtained for higher projective classes.