The person in question is currently staying in America.
Literal
In-question person [topic-は] currently America [at-に] in-the-middle-of-staying [copula-である].
問題の here means 'in question, at issue' — not 'problematic.' 目下 (もっか, 'currently, at present') is a formal time adverb. 滞在中 uses ~中 ('in the middle of') as a suffix on 滞在 ('stay, sojourn').