The tip of my nose is red a month after surgery

I had rhinoplasty on January 4th of this year. I originally had a short nose and it was slightly raised. The day after the surgery, the tip of my nose was red and has been red ever since. The hospital can't tell if it's inflammation or not, but they told me to take antibiotics, and it's already been 28 days. For about 10 days after the surgery, my nose was swollen and sore when I touched it. Now, the swelling has gone down and it's fine when I touch it, but the tip of my nose is still red. Is this inflammation? The hospital keeps telling me to take antibiotics just in case, but I took the medicine for a month and even got a shot for 10 days in between, but it didn't help much. Is this inflammation? The redness isn't that severe, so can it last this long? It's not that it's red, but only the tip of my nose is red. Is there a way to go to a university hospital or something and get tested for inflammation? What kind of tests should I request?