In the Partition problem, there is a set of integers, and the goal is to decide whether it can be partitioned into two sets of equal sum. This problem is known to be NP-complete.
Suppose we are given an instance and we know that it admits an equal-sum partition. Can this partition be found in polynomial time (assuming $P\neq NP$)?
Let's call this problem "GuaranteedPartition". On one hand, apparently one can prove that GuaranteedPartition is NP-hard, by reduction from Partition: if we could solve GuaranteedPartition with $n$ numbers in $T(n)$ steps, where $T(n)$ is a polynomial function, then we could give it as input any instance of Partition and stop it after $T(n)$ steps: if it returns an equal-sum partition the answer is "yes", otherwise the answer is "no".
On the other hand, the GuaranteedPartition is apparently in the class TFNP, whose relation to NP is currently not known.
Which of the above arguments (if any) is correct, and what is known about the GuaranteedPartition problem?