word choice - "Would you have liked to have been" vs. "would you have liked to be"

I was interested in the following sentence which appeared in an article titled “No Rest for the Weary” in The New York Times (February 15, 2008).

Would you have liked to have been president from 1862-1864?

It sounds ungrammatical to my ear as the journalists (MICHELLE SALE and YASMIN CHIN EISENHAUER) did use "to have been" rather than "to be", but I am not able to find what rules govern this problem.

So, I would write:

Would you have liked to be president from 1862-1864?

Am I right? If so, why?