Just what are you looking at when you see VERTICAL movement?
This movement sounds like it is behind the ball/socket connection.

The big end of the inner tie rod attaches to the shaft that comes out rack housing. Correct? You are saying that the ball joint has little or no play.

The VERTICAL movement you speak of is inboard of that. In other words the SHAFT is showing you VERTICAL movement with the ball attached.
If this is the case the shaft is being held by worn out sleeve bushings and the rack must be bad.

If this is the case I am very surprised you are not mentioning any oil leakage. A bunch of oil in the bellows, rubber boots or “gaiters” since we have tipped toed or walked around the vertical movement analysis or descriptions of this.

If this has happen, it is safe to say, that the bellows have been bad in the past, the power steering fluid was never ever changed or this rack has in the neighbor hood of over 300k on it. That is if it was the original rack. If it was not, you do not want that name rebuild again!

Phil

The ball-and-socket part of the tie rod attaches directly to the rack gear, so if the movement is "inboard" of the ball-and-socket, the rack gear itself is moving radially. If so, the bushing in that end of the rack housing is shot.

In the Volvo factory book the rack rebuild process appears to require a number of specialized tools and precise measurements. When I was faced with a leaking rack, I opted for a rebuilt unit. After a bad rack from another rebuilder, I got one from Jorgen (jorgenauto.com), which came with new tie rods. About $200 when you return your old one. It's leak proof, smooth and tight, years later.
--
Bob: Son's XC70, daughter's 940, my 81 and 83 240's, 89 745 (V8) and S90. Also '77 MGB and some old motorcycles

"When I slid the gaiter back, the inner tie rod attaches at a point outboard of where the vertical movement is. There does not appear to be much if any movement in the inner tie rod ball and socket."

Greg,the ball and socket under the 'gaiter',is not the problem.The outer tie rod end is most likely the problem.Do you have a Bentley manual for your car? If not,you should invest in one.It's well worth the $50 bucks.

I had the same type shimmy problem and it was the tie rod end,(outer?).
Jack up the car with both front wheels off the ground and wiggle the tires.
Best wishes...d