NOT A TEACHER
(1) As we say in English, you have stirred up a hornet's nest. That is, your question is really difficult and will cause people to disagree with one another about the "correct" answer.
(2) I, of course, do not claim to have the correct answer. Only the teachers here can give you an answer that you can take to the bank (accept with confidence).
(3) I can, however, share some research that I have done on this matter:
(a) The greatest grammarian ever (in my opinion, of course), Professor George O. Curme, gives this sentence: He was busy two years writing this book.
(i) IF I understand him, he says that "writing this book" is a participle modifying the verb. In other words: He was busy two years [while he] was writing this book.
(b) Then Professor Quirk and his colleagues in their huge grammar give this example
Margaret is busy writing letters.
(a) They call "writing letters" a participle clause.
(b) I think that they feel it modifies the adjective because they title this section in their
book as "Adjective complementation by an -ing participle clause."
The distinguished scholars also point out that we often can use a preposition. Their example:
I am busy (with) getting the house redecorated. They call this a case in which "a preposition occurs between the adjective and the participle clause." They do not say this, but I think that most books feel any -ing word after a preposition must be classified as a gerund. Am I right?
(4) Finally, I found this explanation a few years ago. And it is, I guess, my favorite.
(a) Many (many!) years ago, a- was a preposition in front of gerunds. (It meant something like "on.")
(b) So many (many!) years ago, you might have a sentence like:
She has been busy a-ironing this evening.
(i) Then, over the years, English speakers dropped the "a-" and the sentence
became simply "She has been busy ironing this evening."
I guess that one would analyze it by saying: It's a gerund following a suppressed preposition (that is, the preposition a- has been dropped but it is still there in theory).
(4) Well, now you understand why I said that you stirred up a hornet's nest. It will be fun to see what the teachers say. After you study their answers, then you will have to make a decision as to which explanation you wish to accept.
Student or Learner