Let $x_1, \ldots, x_n$ be **possibly dependent** random variables, each taking values $x_i \in \{0, 1, 2\}$. Suppose further that in every outcome the number of random variables that equal 2 is exactly 1. Now for each $i \in \{1, \ldots, n\}$ define
$$
f_i = \begin{cases}
\Pr[x_i = 2 \mid x_i \geq 1] & \text{if } x_i \geq 1\\
0 & \text{if } x_i =0
\end{cases},
$$
and for each $i \in \{1, \ldots, n\}$ let $y_i$ be a Bernoulli random variable that is 1 independently with probability $f_i$ and 0 otherwise.

Is the following conjecture correct or is there a distribution on $x_i$'s refuting it?

**Conjecture:** There is a fixed $\epsilon > 0$ (i.e. $\epsilon$ being independent of $n$) such that with probability at least $\epsilon$, there is exactly one index $i$ where $y_i = 1$.

Related question: Bounds on variance of sum of dependent random variables