When in doubt... see what the standard says.
IEEE Std 1003.1-2001:
The system shall initialize this variable at the time of login to be a pathname of the user's home directory.
For an implementation that conforms to IEEE Std 1003.1-2001, describes a feature or behavior that is mandatory. An application can rely on the existence of the feature or behavior.
For an application or user, describes a behavior that is mandatory.
Qt expecting $HOME to be set seems in-line with the base standard.