Positive correlation of paths in the annealed random graph model for all parameters

Let $G$ be the annealed randomly oriented Erdős–Rényi graph $\vec G(n,p)$, with distinct vertices $a,s,b,t$, and let $\{s\to a\}$ and $\{t\to b\}$ be the events that directed paths exist from $s$ to $a$ and from $t$ to $b$. Positive-correlation conjecture. For every $n\geq 4$ and $p\in(0,1]$, the events $\{s\to a\}$ and $\{t\to b\}$ are positively correlated. The paper proves this for sufficiently large $n$ at each fixed $p\in(0,1]$ and reports computational support for the stronger uniform assertion, which remains open in the supplied text.