The Mieczkowska–Sileikis conjecture for maximal i.i.d. tail probabilities

Let $m_k(x)$ denote the supremum of $\mathbb{P}(X_1+\cdots+X_k\geq 1)$ over nonnegative, independent and identically distributed random variables with $\mathbb{E}(X_i)\leq x$. For a positive integer $k$, let $x_0(k)$ be the solution of $1-(1-x)^k=(kx)^k$. Mieczkowska–Sileikis conjecture. For every positive integer $k$ and $x\geq 0$,

$$m_k(x)=\begin{cases}1-(1-x)^k & \text{for }x<x_0(k),\\(kx)^k & \text{for }x_0(k)\leq x<1/k,\\1 & \text{for }x\geq 1/k.\end{cases}$$

The paper proves the claim for $k=3$ and all $x$, and for $k\geq5$ when $x<1/(2k-1)$; the remaining cases are not resolved here.