A $2$-branching construction for the $\chi \leq 2r$ bound

The string repetitiveness measures $\chi$ (the size of a smallest suffixient set of a string) and $r$ (the number of runs in the Burrows--Wheeler Transform) are related. Recently, we have shown that the bound $\chi \leq 2r$, proved by Navarro et al., is asymptotically tight as the size $\sigma$ of the alphabet increases, but achieving near-tight ratios for fixed $\sigma > 2$ remained open. We introduce a \emph{2-branching property}: a cyclic string is 2-branching at order~$k$ if every $(k{-}1)$-length substring admits exactly two $k$-length extensions. We show that 2-branching strings of order~$k$ yield closed-form ratios $\chi/r = (2\sigma^{k-1}+1)/(\sigma^{k-1}+4)$. For order~$3$, we give an explicit construction for every $\sigma \geq 2$, narrowing the gap to~$2$ from $O(1/\sigma)$ to $O(1/\sigma^2)$. For $\sigma \in \{3,4\}$, we additionally present order-$5$ instances with ratios exceeding~$1.91$.