Hi,
What you described sounds like a failing crankshaft position sensor. When they begin to fail, oftentimes, they are adversely affected by heat. So, when he shuts the engine, the temp under the hood increases. Once it has a chance to cool, it starts working again.
This is a theory at this point, but it is my first suspect. Take a look through this link and let me know if anything mirrors what you experience:
https://www.2carpros.com/articles/symptoms-of-a-bad-crankshaft-sensor
If you are comfortable doing it, see if the ignition spark is present at the spark plugs when it won't start. If I am correct, there won't be any spark.
Here is a link that explains how that is done:
https://www.2carpros.com/articles/how-to-check-for-ignition-spark
Let me know what you find, if you have questions, or what I can do to help.
Take care,
Joe
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Monday, April 3rd, 2023 AT 5:32 PM