Why is there a minimum altitude for at least some high-speed jet engine restarts?
Generally when you see those kinds of limitations it means that the engine or airframe developer had trouble getting successful, repeatable relights outside of the published envelope during development flight testing and you do see it with other engines (the CF-34 that I am familiar with has a similar but much less extreme limitation).

In that case it is probably related to density/velocity of the air flow into the burner can during windmilling and I would imagine that there were issues with getting a successful lightoff every time, or excessive peak ITT during the start, or maybe surging problems or something like that.

You'd have to talk to a Honeywell person for those details.