Could it be the heater valve?

I have a 2002 325i wagon with an intermittent heat problem. When running a few short errands thus turning the car on and off within 5+ minute intervals, very high heat will come on and stay on even if the A/C is on. Nothing stops the heat meaning lowering the temp, shutting the car off/on, no control adjustments. Once the car has sat for the night, it's fine the next day. Dealer shop has tried to replicate the problem and can't. They have said possibilities will be a very expensive fix of over $1000. I don't want to gamble until the problem is identified. Could it be the heater valve?

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[Nick at Pelican Parts]

The duo valve can get stuck open and deliver hot coolant to the heater core. Try cleaning the valve

- Nick