The nice question below was answered in the affirmative in On the sum of uniform independent random variables

*Let $X_1,...,X_n$ be independent uniform random variables in [0,1] and assume $c>1/2$. Is it true that $$\mathbb{P}\left[\sum_{i=1}^n X_i \leq n \cdot c\right]$$ is increasing with respect to $n$ (for all $n\geq 1$)?*

My question is, can one characterise distributions of $X$ for which this type of monotonicity holds for all $c>\mathbb{E}(X).$

Or maybe it is easier to show for $c$ greater than the median of $X$.

It might be that a nonincreasing probability density is sufficient. But is it necessary?