Samuels' lower-tail product conjecture

Let $n \in \mathbb{Z}_{\ge 1}$, let $\mu_1 \ge \dotsc \ge \mu_n > 0$ satisfy

$$\mu_1+\dotsm+\mu_n=1,$$

and let $X_1,\dotsc,X_n$ be independent non-negative random variables with $EX_i=1$ for every $i$. Define

$$Z=\sum_{i=1}^n\mu_iX_i,$$

and let $T=1+\delta$ with $\delta>0$. Samuels' conjecture.

$$P(Z<T)\ge \min_{1\le i\le n}\prod_{j=1}^i\left(1-\frac{\mu_j}{T-\sum_{k=i+1}^n\mu_k}\right).$$

This is a related conjecture of Samuels concerning the lower tail of a weighted sum of independent non-negative random variables. The source states that Samuels' conjecture implies Feige's conjecture, but provides no resolution of Samuels' conjecture here.