LM, the tether could lay in a channel, but it would have to be a very smooth channel transition... We have always had it just laying on the bottom, all the others folded on top of it, the chute folded in and a spring loaded pilot. If you don't have enough tube, it gets really hard to pack the chute and tether into the tube and get the pilot in there as well. Before you pick a tube, pick a pilot... Then talk to the chute manufacturer to understand how much volume you need for your chute. Diest did all that for us 40 years ago... then the chute grew, and pilots grew and it got hard to pack.... That is why Stan suggested you can't have too much room in the chute tube.
Our liner had chute tubes that were larger in the front than the rear... thought about the same type for this car but that is just one more thing to build.
Incidentally, I found 5 inch schedule 5 304 Stainless at Pittsburgh Steel Products, they have a $150 minimum, but they had a 7 ft drop, so I now have some extra tube coming... Yes I spent more than I wanted but I will have plenty of extra for the new liner, and maybe some for someone else... I have enough for 3.