As I imagine we all know, one can arrange for Boeing to display almost anything in any location for a price. However, as has been stated, TAT is the critical factor in ice accretion, i.e., the TAT must be below a specific threshold for supercooled water to freeze on impingement. This is what drives the "turn the ice protection on" logic, as well as the strategy of high speed = high TAT = no ice.
That said, SAT is strongly influential in whether there is any supercooled water at all. Statistically, when the SAT is below -40C, there is almost no supercooled water present. This is what drives one leg of the "turn the ice protection off" logic, the other being TAT. The idea is that below -40C, there just isn't any supercooled water, so one doesn't need the ice protection.
However, as has been mentioned on a number of other threads, it is very possible to encounter supercooled water at a SAT of less than -40C when in the vicinity of large cumulus buildups, since the vertical motion can lift a water droplet rapidly into a SAT that is very cold. So while the statistical model of supercooled water and SAT is pretty valid, it isn't perfect.
Finally, none of this addresses the relatively newly understood issue of ice crystals, which are not part of the SAT less then -40C model at all, and seem to have had some success in instigating rollbacks, etc.