Presumably the vertical position of the book gutter will vary depending on the thickness of the book, and probably how far through the book has been reached? A mechanical microswitch might be one easy, readily available and inexpensive solution?
If the gutter position does not in fact vary, a microswitch would certainly a good starting point.
There are also, I think, photo-optical sensors that might provide a non-contact solution, some research needed there, but this is one that can be used as a limit switch might be usable, the limit distance would probably vary with the paper reflectivity, but that probably shouldn't be a serious issue. Another type might use a light source emitting at an angle and a light detector also set at an angle.
Are there no clues from close examination of the video? Probably little chance of a Japanese speaker reading the thread in the near future, but you could always try contacting Moose...