Jars are always "best practice," but I have run the experiment, and if you are actually smoking from the tub, not aging it, it does quite well. I don't detect any off-gassing of the plastic into the blend, and the moisture level remains good. To make it clear, I have that same blend, Hearth and Home, Midtown series, Chestnut. If you want to be fastidious, jar it, but if you smoke it from the tub, in my experience, it will be okay. For the sake of space, I eventually transferred it to a jar, but I am using the tub for another blend I use regularly, properly relabeled.
P.S. I don't think Chestnut is a Cornell & Diehl product. It seems mostly sold through Pipes and Cigars.