Sounds like a cough to me. My first thought would be kennel cough, but there are a lot of reasons for coughing. It would be good for the vet to see the video. Sounds like they are treating it as an infection just in case. Antibiotics will really only help secondary infections though. If it is an infection, it could last a long time. Most vets probably won't give antibiotics for weeks unless there is a secondary infection.
When my dogshad kennel cough, they were put on an AB only after they developed the nasally symptoms. They were on it for about ten days.
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