Hartke's integrality gap conjecture for trees with at most 12 vertices

Let $T$ be a tree, and consider the linear program obtained from the MacGillivray–Wang integer program for virus spread by relaxing $x(v)\in\{0,1\}$ to a linear constraint and adding constraint (6):

$$\sum_{v \succeq u} x(v)+\sum_{u \succeq v,\,l(v)=i}x(v)\leq 1,$$

for each vertex $u$ and $i>l(u)$. Hartke's integrality gap conjecture. The tree in Figure~ is the smallest tree such that the LP optimum under constraint (6) differs from the integer-program optimum. Equivalently, the strengthened linear program realizes the integer-program optimum for every tree with at most 12 vertices. The conjecture concerns the occurrence of integrality gaps: the displayed 13-vertex tree has LP optimum $m^*=7.5$ and integer-program optimum $m=7$, while the claim asserts that no smaller tree has such a gap.