Will this tube accelerate?

My question relates to this question and more specifically an answer (by ‘oliver’) it received.

A sealed tube is heated at one end and cooled at the opposite end. The OP wanted to know whether there’s a pressure gradient in the tube.

The question itself doesn’t particularly interest me but how ‘oliver’ set out proving there’s no such gradient does. Basically he contends that:

If the pressure was higher on the hot end than on the cold end, the
total force would also be higher on the hot end than on the cold end,
and hence, the container would accelerate without external directed
force being applied, which would contradict momentum conservation.

So he concludes that no acceleration means no pressure gradient ($P_H=P_C$).

I find that hard to comprehend. There are no external forces and no changes of momentum (Ok, maybe just a little at the start of the heating/cooling)

It reminds me of a man in a small sloop (dinghy). The man pushes on the bow ($F_B$) and the stern ($F_S$). If $F_B>F_S$ would the sloop move? No, of course not.

So how can we prove ‘oliver’ is right (or wrong)?