Not indefinitely. But the question we should ask is, do we benefit as a society by removing these people from unemployment? Will it affect our deficit and debt in a significant manner, one way or the other? Is it worth it?
My impression is that it is not worth it. We gain nothing significant by cutting these people off at this time. They will add to the people looking for work, or to the homeless. They will stop being able to pay their bills, which will in turn cause companies to fail. In return for this, we subtract a few millions from a deficit which is close to a trillion a year.
Makes no sense.

or maybe they will go out and get a job even if it is outside their career path..
I've mentioned this before but I was laid off from my first job out of college. While my unemployment checks were coming in I was out daily hunting for a new job in my career field .. This was the "ancient times" before internet where you had to check the newspapers, and cold call companies and mail resumes.
When I received the warning that my unemployment checks would end in a month I started considering other possibilities, including going back to school and get my bachelor degree.
Once the checks ended I got a full time job at a Restaurant as a Prep Cook, started looking into college, and continued to send out resumes.. about a month and half later I got a job in my career path and as they say, the rest is history..
To this day I still think being forced to re-evaluate my career and start working Full time made me look , and feel, less "desperate" at the interview and helped me land the job.