This is dumb
That rumor is obviously unverified. Makes no sense that the league would sign off on allowing Kylington to be on Injured Reserve for a family illness. When things like this happen - pregnancy, death in the family, etc., - players usually get a short leave of absence and that's get it. They don't get to stay away for months.
They're not putting Kylington or Stone in LTIR because the plan is to accrue cap space for a deadline add. LTIR relief does not accrue.
If Treliving randomly decided to trade Kylington, he'd immediately go out and give up more assets for another defenseman because that's just how he is. In other words, it's pointless.
Fans are impatient but very likely nothing significant happens until closer to the deadline. At that point, depending on the standings, Treliving will either stand pat, sign Coronato when his seasons done, trade the 2023 1st and a B prospect for top rental, or trade one of the big four prospects and a 2nd for a top rental.