The revised SIAM Review conjecture on extremal distributions

Let $X_1,\ldots,X_n$ be iid random variables supported on $[0,1]$, with mean $m$ satisfying $0<m<1$, and write $S_n=X_1+\cdots+X_n$. Define

$$p_n(m,t)=\sup\mathbb P(S_n\leq t).$$

For $n\geq2$ and $0\leq t<mn$, call a distribution extremal if it attains this supremum. Revised SIAM Review conjecture. For any $n\geq2$, $0<m<1$, and $0\leq t<mn$, all distributions attaining the supremum $p_n(m,t)$ belong to the collections described below.

The surrounding discussion proposes collections of extremal distributions with at most three support points, including binary candidates and distributions supported on configurations such as $\{0,t,1\}$. The supplied text does not provide a resolution of this revised conjecture.