Feige's lower-tail probability 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 $M=\max_{1\le i\le n}\mu_i=\mu_1$, $\delta>0$, and $T=1+\delta$. Feige's conjecture.

$$P(Z<T)\ge \min\left(\frac{\delta}{\delta+M},\frac{1}{e}\right).$$

Feige proved a weaker version with $1/13$ in place of $1/e$; the stated constant is sharp in general, but the conjecture itself is presented here as an unproved conjecture.