Optimizing stakes in simultaneous bets

We want to find the convex combination $S$ of iid Bernoulli random variables that maximizes $\mathbf{\text{P}}(S\ge t)$ for a given threshold~$t$. Cs'oka conjectured that such an $S$ is an average if $t\ge p$, where $p$ is the success probability of the Bernoulli random variables. We prove this conjecture for a range of $p$ and $t$.