Stabilizing fuel is something of preference, that in my opinion and experience is over-rated.
Around here we reverse store our snowmobiles in the summer months, putting them away in the spring until winter. We have a seldom used rig that is over 15 years old, used only for breaking a trail these days. It is a 2-cylinder, 2-stroke, oil injected and I run it on regular gas that now has ethanol included. Last spring, as usual, I put it away by just draining the carbs (shut off gas supply and ran till dry) and disconnected the battery in case there was a drain. As usual I got it ready and went and started it up just the other day. I didn't add any gas as the tank was still about 1/4 full, but I did charge-up the battery for a day on the trickle-minder. Turned it over a few times with no response as the gas probably wasn't through yet, so I pulled the plugs and dropped a bit of fresh fuel into the holes for the cylinders. Re-installed the plugs and it fired right up! Smoked a lot as it warmed-up (normal for this 2-stroke) so I thought I'd move it out of the garage to do a bit of a run on the barely snow-covered trails, only to find the belt engagement frozen and not engaging. So I stopped it and greased the mechanism a bit, wriggled it to "unstick" it, and started it right up again. Had a nice ride in the bush and parked it, adding a bit more "fresh" fuel.
Did I say anything about adding any stabilizer?
Nope.