I was waiting for someone with an opinion.  DuckDuckGo:
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Motor Vehicle Maintenance & Repair Stack Exchange
Can coasting in neutral damage an automatic transmission?
Can coasting in neutral damage an automatic transmission?
It is fine to coast in an automatic in neutral as long as the engine is running. If you stop the engine completely and coast (as is often the case when being towed), you can damage the transmission.
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Moreover:
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AutoGuru
Daihatsu Cuore Automatic Transmission Wont Go Into Neutral - AutoGuru
Daihatsu Cuore Auto transmission wont go into neutral Why doesn't my Daihatsu Cuore automatic transmission go into neutral? There are various reasons why your Daihatsu Cuore transmission might have trouble going into neutral: Shift lock solenoid - The shift lock solenoid is the reason why you can't
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Vehicle Freak
Why Is My Car Gear Shift Stuck In Neutral? (Causes And Fixes)
There are a number of reasons why your car's gear shift may become locked in neutral, and we're going to help you identify and fix them. The reasons why your car gear shift is stuck in neutral are: low transmission fluid levels, damaged or frayed shifter cable, TCM problems, or mechanically damaged
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So you do have a shift lock solenoid FWIW.
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