You have to try. If they die, they won't be able to get into Shoe Heaven without a sole!
I had this problem the other week. My old shoes had one sole hanging off around the front to the point that the unused SPD plate fell out.
Mainly because it was all I had, and partly because I thought the very rigid road shoe soles don't bend, I tried 30 minute two part epoxy. I use this stuff to glue handle scales to knives so I know it works.
I weighted it down for an hour and tried the shoes the next day on the rollers. No issues.
Now two weeks later, they are still fine. The soles/glue interface is still perfect.
Like I said, normally with everyday shoes, I know epoxy wouldn't work as it can't flex. With rigid shoes you tend not to walk around in, so far for me, it has been successful.