The MVB conjecture on asymptotically optimal branching variables

An MVB instance consists of variables $(l_i,r_i)_{1\leq i\leq n}$, with recurrence

$$t(G)=\begin{cases}1&\operatorname{if }G\leq 0,\\1+\min\limits_{1\leq i\leq n}\bigl(t(G-l_i)+t(G-r_i)\bigr)&\operatorname{if }G>0.\end{cases}$$

Its variable ratios are denoted by $\varphi_i$. The MVB conjecture. For each instance of MVB, there exists a gap $H$ such that for all gaps greater than $H$, variable $i=\operatorname*{\arg\,\min}_j\varphi_j$ is always optimal to branch on at the root node. The conjecture was introduced after the result that the MVB ratio is $\varphi=\min_{1\leq i\leq n}\varphi_i$. It is false in general, as shown in this paper.