Asymptotic best-variable branching conjecture for MVB

Let an MVB instance have variables indexed by $i=1,\dots,n$, and let $\varphi_i$ be the ratio associated with variable $i$; write

$$i=\operatorname*{\arg\\,\min}_j \varphi_j.$$

Asymptotic best-variable branching conjecture. For each instance of MVB, there exists a gap $H$ such that, for all gaps greater than $H$, variable $i$ is always branched on at the root node.

The conjecture asserts eventual stabilization of the root branching choice at the variable with the smallest asymptotic growth ratio. The surrounding result identifies the MVB growth ratio with the best single-variable branching ratio, but the eventual root-choice assertion is left as a conjecture.