Is it different from the behavior below?
If the operation is the same as the following, it is a specification to keep BEAT. (normal operation)

Hello. I have brand new CDJ3000s, I have updated to the newest firmware. I have issues with the beat jump and hot cues causing the track to jump about a 1/16 of a beat when doing rapid presses of either. It doesn't happen constantly, but it is extremely annoying in the middle of a mix. I use the cues more often than the beat jump for sampling and phrasing. Sometimes it happens with just a single press to jump to a cue, no rapid succession. It happened before I updated the firmware (which seemed to fix the issue) and now it's happening again. I have tried different USB sticks, formatting them completely (not quick format). I am using MP3s currently. I will try them with WAV files and see if the problem goes away. Can you shed any light on why this is happening with what is supposed to be top-of-the-line equipment?
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Is it different from the behavior below?
If the operation is the same as the following, it is a specification to keep BEAT. (normal operation)

I remember here was one other guy complaining about mix between two CDJ 3000s got out of sync when using hot cues (with quantize of course).
I am not sure if I understand the description from Tomo correctly, but my understanding is:
Even if quantize is set to some higher value, e.g. 1 beat, if you press the HotCue button within the first 1/32 of the new beat, it will jump to the HotCue immediately anyway (without this, it would wait for the next beat to begin because quantize is set to 1 beat). This is to allow some "unprecise" hit of the HotCue button to trigger it immediately (even with quantize on).
However, as can be seen from the diagram, the CDJ should take into account the time already played from beat 2 and align the HotCue playback accordingly (skip the blue part from the HotCue) to keep everything in sync.
So the test case would be to test exactly this behavior: Hit the hot cue very short (but not exactly) after a beat and see if the mix gets out of sync.