I had a similar problem on our HP C7250. For this model, the gear is very accessible and trivial to inspect and fix with a drop of superglue. 5 minute fix!
The shaft with the upper 4 grey paper rollers was not turning reliable because the plastic gear was slipping on the shaft. The shaft had slide to the side and the spline on the metal shaft was not fulling inserted into the gear.
The fix is to apply a tiny drop of superglue to the shaft and slide it ‘left’ to fully engage with the gear.
I suspect that this occur when someone aggressively cleared a paper jam at the back of the printer and didn’t pull the paper evenly.
Here’s the location of the gear (see left side of image)
Here’s the ‘before’ photo (see the shiny metal shaft spline to the right of the gear):
Here’s what it looks like after the fix:
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