Yes, this counts as a cast.
Is the qualification a rule that they can qualify later in the engagement campaign? If not, the leads that did not qualify to the receive the current cast can be move to the next cast without the wait. In the Admin section, under your Tags --> Channels, you need to add a stage like "Exclude" which should be the last stage. And when the leads do not pass the qualification, change the program status to "Exclude".