I spoke with the owner last night, the unredacted F4 doesn't show any restrictions. The gun in question was purchased with two other Thompsons all of which are supposedly transferable. While I have little doubt the other two are, this one I'm much less confident about. That concern is further validated by the fact that the previous owner was an ex FFL/SOT so it definitely leaves to door open to the pre sample possibility.
From the sounds of it, they have a copy of the previous transfer form for this gun, he was going to dig that up and start by taking a closer look at it (didn't recall if it was F3 or F4).
I'll be in their neighborhood next Monday, happy to swing in and have a closer look at what's he's found by then, will PM you my contact info.