The additive 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 $a=1$. Additive conjecture. For every $k\geq 4$, if

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

then

$$\operatorname{ex}_\Sigma(s,q)=1+\frac{k-1}{3k+1}.$$

At the endpoint $q=\Sigma_s(K_{1,\infty})$,

$$\operatorname{ex}_\Sigma(s,q)=1+\frac13.$$

The conjecture concerns additive extremal densities over intervals of $q$ of quadratic length not covered by the paper's results. The listed graph patterns provide extremal constructions in the cases $k=4$ and $k=5$, and at the endpoint.