Kenmore ice maker leaking water

Hello,

I have a Kenmore side by side (model # 596.51673100) and the ice maker has recently started leaking into the ice tray. The leaking starts right after water fills into the unit and drips slowly into the tray. The ice maker does not leak continuously, only soon after the fill cycle.

I tried replacing the secondary water inlet valve, but this did not solve the problem. Any other ideas?

Thanks,
Blaine