Let $n,t$ be positive integers and $p_1,p_2,\ldots,p_n$ positive numbers summing to 1. Conjecture: $$ \sum_{i=1}^n p_i (1-p_i)^t \le \frac{n(1-1/n)^n}{t} $$ always holds.

The motivation comes from my missing mass question; the quantity $\sum_{i=1}^n p_i (1-p_i)^t$ is precisely the expected unseen mass after $t$ draws from the distribution $(p_i)$ on $n$ objects.