Parity of Numbers in Amicable Pair

Open Question
It is not known whether it is possible for the numbers in an amicable pair to have opposite parity.

All known amicable pairs consist of numbers which are either both odd or both even.

Progress
If there do exist an amicable pair whose numbers are of opposite parity, then:


 * the even number is either square or $2$ times a square


 * the odd number is square.