This is a game bug that has been around since the beginning. If the miners hat has a decent amount of fuel in it, right click to add instead of dragging and dropping. Doing so will often delete the extra that would normally stay inside your inventory.
You can also take the fuel out, restack and replace inside the hat to avoid this from happening.
I highly recommend you two make a reasonable effort at an appeal (on separate threads) if you're truly adamant about having your bans lifted. I will not be a deciding factor, and I cannot guarantee your bans will be lifted, but just asking to be unbanned is not going to cut it.
Take responsibility and admit your fault and develop an argument on how you'll improve your behavior to align with our community standards.
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