Here's why I don't think that's likely.
1. Stone returns from injury, plays amazeballs and by the time contract talks are allowed to begin he's now an insanely hot commodity and will see huge dollars at UFA. Even if we ultimately want to sign him in the summer, Stone skipping out on his first chance to test the market strikes me as unlikely in the scenario where he's playing well. If nothing else he loses the chance to get other teams to bid up the Coyotes offer.
2. Stone returns from injury and is mediocre through January. Now do you even want to sign him to a long term contract? Remember we already have Murphy and GoGo on term already and OEL will need to be paid here at some point. Even if you did a lot of teams would still throw big dollars/term at him thinking he's still recovering. Stone's best option in this scenario is to wait and see if his performance improves as the year goes on looking at the payday. If he doesn't improve it won't be any problem to get a 1-2 year prove it deal from the Coyotes or somebody else. There is still not a great incentive for him to sign early.
3. Stone returns and is just outright awful, looks like he's lost a step despite claiming to be 100%. Yay, we can extend a possibly broken defenseman to a short term deal!
This isn't even getting into the issues of expansion protection and the opportunity cost of not trading him for assets.