I have WQ-T352E double cassete player with the same issue
The provided link below contains the service manual and everything you need to know.
But if you get stuck. Don't hesitate to ask.
For the life of me I can't find how the autostop is triggered
In general the automatic stop can be triggered
A number of ways (This is only an truncated synopsis) electronically using sensors that will detect the physical movement of the tape in motion and or lack of motion, optical or hall effect, electromagnetic most commonly tied with the tape rotational counter and of course mechanically incorporating but not limited to capstan, pinch roller, supply reel, take up reel
Fly wheels and physical levers with associated pins. Tension,pressure, inertia, various cogs and sprockets working in unison, obviously under normal conditions so your tape doesn't get stretched preventing high frequency response degradation, play back, record , wow & flutter.
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