Insertions into a max heap with atmost one swap

Question

Is there a algorithm to perform insertions into a heap with atmost one swap (O(log n) comparisons are allowed)

Answer 1

No.

Consider this heap:

Suppose you add 200. Obviously it will have to become the new root.

So where does 100 go? It can't become a child of 3, and that's what it would have to do if you only have one swap.