I have tested the battery and alternator and both test fine.
Make sure you have good battery cables and battery terminals as well. Check for any fine cracks on the terminals, which can lead to a loose connection. A faulted sensor like a coolant sensor or air temperature sensor can offer false readings to the ECU and in turn the ECU will send the wrong air-fuel mixture to the engine causing it to start hard or run poorly.
Newer vehicles have a voltage regulator that is integral to the alternator assembly. If the voltage problem is affecting the whole car, this may be the problem. To have all these items checked, I recommend enlisting the help of a qualified mechanic that can inspect the hard start, pinpoint any faults, and perform any repairs that may be needed.
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