IMO I don't think the 97 XP is setup very well from the factory. I had the same ski, & I messed with the carbs so many times trying to resolve hard warm restarts. I tried different springs / pop off pressures. Finally after many failed attempts I decided to go outside the specs. What I found on mine was the low speed circuit was too lean.
When I finally figured it out, I ended up with my low speeds out well over 2 turns, maybe 2.5 or so? I had it running on the trailer at 3K & kept backing them out. When I got the screws out far enough, the idle started to drop as it finally started to get more to the rich side than it ever had before. I then had to turn up the idle speed (throttle plates) to get it back to 3K on the trailer. The ski never ran better after that. Restarts were nearly instant warm. the combination of those two simple things made a huge difference for me.
If I remember correctly the 97 XP jets were a one year only. There's something about those 97's thats not quite right IMO.
I will search for the thread I made at the time. I would suggest you try it. It costs nothing to do, & you don't have to take apart anything to do it. Unless you currently have other things changed too far from spec.
Not saying yours is lean on the LS, but its not uncommon on the 97 XP either.