The current description of the "sonar.cpd.cross_project" property is misleading and can lead users to think that when set to false (which is the default), duplications will only be detected inside a project, including its (potential) sub-projects. Whereas when set to false, duplications are only detected inside a sub-project.
So the description should be updated to:
By default, SonarQube detects duplications at sub-project level. This means that a block duplicated on two sub-projects of the same project won't be reported. Setting this parameter to "true" allows to detect duplicates across sub-projects and more generally across projects. Note that activating this property will slightly increase each SonarQube analysis time.