Main contact K10 wont close till you pull down the tiller. But if you have a C28 all truck functions are suspended, so you will not get K10 to close. K4 should be open during C28 and yes if K4 is closed K10 should be closed as well. Are you testing across contacts or measuring voltage across the relay coil to determine K4's status?
From what you just said, if the battery is falling down to 14V when powering up the unit then you probably have a bad cell or an overall tired battery pack.
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