Henning's Sweller-start competition-independence conjecture for trees

Let $T$ be a tree of order $n$. The competition-independence number $I_s(T)$, namely the length of the competition-independence game when Sweller moves first and both players play optimally, satisfies

$$I_s(T) \geq \frac{3}{7}n.$$

Henning's conjecture. For every tree $T$ of order $n$, $I_s(T) \geq \frac{3}{7}n$. The paper disproves this conjecture by exhibiting arbitrarily large trees for which the Sweller-start game lasts at most $(5n+26)/12$ moves, while proving the general lower bound $I_s(T)\geq(5n+3)/13$.