Engine jumping out of 4th 1000J

Under hard acceleration in 4th and with slight reduction in throttle it will want to jump out of gear into a false neutral. I'm thinking the detent spring is weak. Any other suggestions?

Sound like shirt forks to me.
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Behind the sprocket cover is a tranny cover. Behind that are the shift return springs, shift arm, etc. Kawasaki installed a locking cam on the later engines. It's possible the spring or springs could be weak or broken. I would check that first before deciding to split the cases.
Steve

Springs don't just weaken with mileage, but time also.

Want to look at the barrel rollers while you're in there with the cover off.

If springs and barrel aren't the issue, a bent/worn fork could do it too.

All that eliminated, you can drop the oil pan and get a look at the gear dogs. Significantly rounded off dogs will produce popping out or gear. Although it's typically a 2nd gear issue on a street bikes.

I had a problem a long time ago where it would pop out of gear and it was the groove for the lock washer(26) on my shift rod(27) was cut too shallow and the lock washer came off and the rod slipped out, thankfully not damaging anything.

Don't ride with it like that!