$JENKINS_ARGS populated by /etc/default/jenkins and used by the init script doesn't contain "-prefix" option by default. Adding "-prefix=$PREFIX" to JENKINS_ARGS in /etc/default/jenkins is preventing the package to be upgraded automatically (by cron-apt) since a conflict which dpkg couldn't resolve, leaving it in broken state, without running Jenkins daemon.

Instead, the init script could use $PREFIX (if any) directly to construct a "--prefix=$PREFIX" part of the start command.