Polynomial-time indistinguishability conjecture for planted and unplanted number partitioning

Let p-\texttt{NPP} and u-\texttt{NPP} denote the planted and unplanted random number partitioning problems, respectively. For a fixed $c>1$, consider distinguishing these two models in the regime described by the hypothesis-testing problem.

Polynomial-time indistinguishability conjecture. Fix any $c>1$. There exists no polynomial-time algorithm that distinguishes the p-\texttt{NPP} from the u-\texttt{NPP} with probability greater than $\frac12$.

This conjecture concerns the computational tractability of hypothesis testing between the planted and unplanted models. The source motivates it by noting statistical distinguishability when values below $2^{-cn}$ occur, while leaving efficient distinguishing open.