Samuels's conjecture on sums of independent nonnegative random variables

Let $0\le\mu_1\le\cdots\le\mu_l$ with $\sum_{i=1}^l\mu_i<1$. Define

$$P(\mu_1,\ldots,\mu_l)=\inf\mathbb{P}(X_1+\cdots+X_l<1),$$

where the infimum is over independent nonnegative random variables $X_i$ with expectations $\mathbb{E}X_i=\mu_i$, and define

$$Q_t(\mu_1,\ldots,\mu_l)=\prod_{i=t+1}^l\left(1-\frac{\mu_i}{1-\sum_{j=1}^t\mu_j}\right),\qquad 0\le t<l.$$

Samuels's conjecture. For all admissible values of $\mu_1,\ldots,\mu_l$,

$$P(\mu_1,\ldots,\mu_l)=\min_{t=0,\ldots,l-1}Q_t(\mu_1,\ldots,\mu_l).$$

This probabilistic conjecture underlies the paper's treatment of fractional matching thresholds; the source does not state a general resolution, so it remains open here.