The number of correct guesses with partial feedback

We consider the following game. A deck with $m$ copies of each of $n$ distinct cards is shuffled in a perfectly random way. The Guesser sequentially guesses the card from top to bottom. After each guess, the Guesser is informed whether the guess is correct. The goal is to maximize the expected number of correct guesses. We prove that, if $n= \Omega(\sqrt{m})$, then at most $m+O(\sqrt{m})$ cards can be guessed correctly. Our result matches a lower bound of the maximal expected payoff by Diaconis, Graham and Spiro when $n=\Omega(m)$.