On the Probability a Weighted Bernoulli Sum Exceeds Its Mean

Let $w_1, \dots, w_m$ be positive real weights whose sum is $1$, and let $v_1, \dots, v_m$ be i.i.d. Bernoulli$(p)$ random variables. If we let $X=\sum_{i=1}^m w_i v_i$, then we conjecture that for all $0\leq p\leq 1/3$ we have $$\mathbb{P}\big[X\geq \mathbb{E}[X]\big]\geq p.$$ In this short note, we observe a connection of this conjecture with a version of the Manickam-Mikl\'os-Singhi conjecture, which allows one to prove it for sufficiently small values of $p$.