Berikkyzy–Hogenson–Kirsch–McDonald conjecture on extremal star counts

Let $G$ be an $n$-vertex graph satisfying one of the conditions $(H_1)$–$(H_5)$ in Theorem 2, and let $\textstyle\ell=0,-1,1,k,k$ correspond to those cases, respectively. For positive integers $t$ and the associated quantities $g_t(n,\ell,1)$ and $g_t(n,\ell,i_0)$, assume

$$t\leq \left\lfloor\frac{n+\ell}{2}\right\rfloor.$$

Berikkyzy–Hogenson–Kirsch–McDonald conjecture. If $t\leq \left\lfloor\frac{n+\ell}{2}\right\rfloor$, then, for sufficiently large $n$,

$$g_t(n,\ell,1)\geq g_t(n,\ell,i_0).$$

This conjecture identifies which of the two candidate extremal graph families gives the larger number of $t$-vertex stars in the lower-half range of $t$. The preceding results establish the opposite comparison when $t\geq (n+\ell+1)/2$, while the asserted inequality for sufficiently large $n$ remains open.