Another alternative which is correct, "This conversation slows down completion of my work." The word "the" is not required here.
And another alternative, "This conversation slows the completion of my work.", should also be accepted. "Slows" and "slows down" are equivalent, "down" is not required.
The official answer is not good English and would never be said by a native speaker. A more natural version would be: This conversation is delaying me finishing my work