The multiplicative conjecture for star-pattern intervals

Let $s\in\mathbb{Z}_{\geq 4}$, and let $G=G^{(a)}$ be any graph pattern in the statement, with ambient edge multiplicity $a=2$. Multiplicative conjecture. If

$$\Sigma_s(K_{1,4})\leq q<\min\left(\Sigma_s(K_{1,7}),\Sigma_s(K_{1,\infty})\right),$$

then

$$\operatorname{ex}_\Pi(s,q)=e^{\pi_{\operatorname{Petersen}}}.$$

For all $k\geq 7$, if

$$\Sigma_s(K_{1,k})\leq q<\min\left(\Sigma_s(K_{1,k+1}),\Sigma_s(K_{1,\infty})\right),$$

then

$$\operatorname{ex}_\Pi(s,q)=e^{\pi_{K_{1,k}}}.$$

Finally, at $q=\Sigma_s(K_{1,\infty})$,

$$\operatorname{ex}_\Pi(s,q)=e^{\pi_{K_{1,\infty}}}.$$

The conjecture proposes the multiplicative extremal values on the remaining star-pattern intervals for ambient multiplicity $2$. The paper gives corresponding extremal constructions, but the full assertions are presented as tentative conjectures.