***** NOT A TEACHER *****
Great question, Elitez:
First, let's do what my teachers taught me when analyzing a sentence: simplify it.
Let's work on "A softer attitude can reduce the intensity of confrontations, especially on controversial issues."
Do you see what I think I see?
If I understand Professor Quirk's huge book, "especially [confrontations] on conversial issues" is a noun phrase that is in apposition with "confrontations."
Professor Quirk gives this sentence: "We want to invite a number of friends, especially Joan and Betty."
He (and his colleagues) say that "an explicit indicator" [such as "especially"] shows "that the particularization [pointing to Joan and Betty] has been chosen because it is in some way prominent." [That is, "We" really, really want Joan and Betty to come to our party above all our other friends.]
By the way, let's change your sentence to the passive.
We get: "The intensity of confrontations, especially [confrontations] on controversial issues, can be reduced by a softer attitude."
In my opinion (opinion), it seems pretty clear that "especially [confrontations] on controversial issues" is in apposition [a relationship] with "confrontations."
Finally, here's a sentence that I copied in my notebook several years. (Sorry that I did not note the source): "I asked him about the discrepancy [difference] between his approach and the White House's [approach], especially on the soda tax."
In my opinion, "especially on the soda tax" is in apposition with "discrepancy."
Let's try the passive "test": "The discrepancy, especially [the discrepancy] on the soda tax, between his approach and the White House's was asked of him by me."
Credit: IF my idea is essentially correct, most credit goes to Quirk and his three colleagues in A Comprehensive Grammar of the English Language (1985), page 1316. Credit also goes to a result on the Web (which I forgot to write down) that a prepositional phrase can be an appositive because there is an understood subject. Fof example, "especially on controversial topics" must have the understood noun "confrontations" as the subject. So "especially [confrontations] on controversial topics" is, therefore, a noun phrase in apposition with the "confrontations" that actually appears in your sentence.