Hurlbert–Kamat leaf-centred maximum-star conjecture for trees

Let $T$ be a tree, let $k\geq 1$, and let $\mathcal{I}_{T}^{k}(v)$ denote the family of independent sets of size $k$ in $T$ containing $v$. A leaf-centred maximum-star conjecture. There exists a leaf $\ell$ of $T$ such that

$$|\mathcal{I}_{T}^{k}(v)|\leq|\mathcal{I}_{T}^{k}(\ell)|$$

for every $v\in V(T)$. This conjecture was disproved by independent work of Baber and Borg, which produced counterexamples for every $k\geq 5$.