Being by nature a bit skeptical, I decided to try to replicate this experience myself. After work, I went to the store and got a box of the same ice cream sandwiches.
For those of you interested, here is the ingredients label.
When I got home from the store, I put the box of sandwiches in the freezer for approximately an hour, to refreeze them from my trip home. Once they had hardened back up, I took one sandwich and set it outside. Notice that the thermometer is showing mid 70s.
Also note that the sky was very overcast. Yes, the reporter had a sunny day, but this is not a factor I could control.
After 5 minutes, you can clearly see the ice cream is starting to soften and melt around the corners.
And the temperature is still showing mid 70s.
After 15 minutes, even more melting has occurred.
And the temperature is holding in the low to mid 70s. I suspect the slight drop is due to the proximity of the frozen treat.
After 20 minutes, there isn't much left.
By 25 minutes, the ice cream has almost completely melted.
And after 30 minutes, it's gone.
Notice that the temperature has remained stable in the low 70s.
In addition to wasting a perfectly good ice cream sandwich, I think I can conclusively say that this myth has been busted. If my sandwich has completely melted in less than half an hour, on an overcast 70 degree day, I don't find it plausible at all that one can remain solid after 12 hours on an 80 degree day, and much less so overnight.
My guess is it's just somebody looking for their 15 minutes of fame.