Gas Dryer won't cycle burner after start up

I've got a Whirlpool gas dryer (Model #LGR764AQO) that heats up just fine when you first turn it on. A few minutes after the burner has cycled off I can see it trying to re-ignite because the igniter glows, but no gas cycles on this time. What is the most likely cause?
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CraigT wrote:

My guess is the valve coil(s). It may work OK when cold, bot it becomes less reliable as it heats up. You can get a new coil set for a little over $20 from www.repairclinic.com.
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