Existence of Coprime Amicable Pair

Open Question
It is not known whether it is possible for the numbers in an amicable pair to be coprime.

All known amicable pairs share at least one common prime factor.

Progress
If there do exist an amicable pair whose numbers are coprime, then, according to, their product exceeds $10^{67}$.

Such numbers cannot be created by Thabit's Rule, or any other similar construction for that matter.