Dwarf, thanks. Interesting reading. My first impule is to discount this idea because it would apparently have a lot of mechanical complexity, all of which would add a lot of both weight and cost to the design. The sound reduction part of the proposal is very doable though, and something that Cntry, myself, and some others have looked into already.
I think an effective shroud can be built, and at reasonable cost. My thinking is that the forward hoop of the shroud (duct) needs to be slightly smaller in diameter than the aft hoop so that the shock waves (sound) generated off of the prop tips are forced to exit to the rear and are prevented from reverberating around in the structure of the cage/shroud itself. The angle would only have to be slight .... maybe something like a 4" difference between those two, with the prop situated somewhat closer to the forward hoop. Enough to reflect the sound aft, but not enough to significantly lower the air velocity through it. The effective slope of a shroud built this way would only amount to a 2" difference on either side.
Recent discussions on here about horizontally mounting the radiator, etc. are also important to this discussion because if a prop is shrouded the inflow of air to the prop will become as managed as the sound and the thrust it produces.
Good post.