Exponential-intersection-tail conjecture for finite transitive graphs

Let $G$ be a finite transitive graph. Say that $G$ has the $EIT(c)$ property, for $0<c<1$, if for every pair of vertices $u,v\in G$ there is a set of paths from $v$ to $u$ equipped with a measure such that two independently sampled paths have more than $k$ intersections with probability decaying faster than $c^k$. Exponential-intersection-tail conjecture. There is $0<c<1$ such that, if

$$\operatorname{diameter}(G)<|G|^{1/3},$$

then $G$ has the $EIT(c)$ property. Such a result would extend the known logarithmic-diameter case, but the stated $|G|^{1/3}$ criterion is open.