Exact rainbow-extremal sets for restricted sum-free colorings

Let $[n]=\{1,\ldots,n\}$. For a subset $A\subseteq[n]$, let $g(A,r)$ denote the number of rainbow sum-free $r$-colorings of $A$, and let $g(n,r)=\max_{A\subseteq[n]}g(A,r)$. A set attaining this maximum is called a rainbow $r$-extremal set. For even $n$, define

$$I_1=\left[\frac{n}{2}-1,n\right],\qquad I_2=\left[\frac{n}{2},n\right],$$

and for odd $n$ define

$$I_3=\left[\frac{n-1}{2},n\right].$$

Exact-extremal-set conjecture. Let $n,r$ be positive integers with $r\geq4$. If $n$ is even and $r\leq7$, then

$$g(n,r)=r^{n/2}\left(3-\frac{2}{r}\right)^2,$$

and $I_1$ is the unique rainbow $r$-extremal set. If $n$ is even and $r\geq8$, then $g(n,r)=r^{n/2+1}$, and $I_2$ is the unique rainbow $r$-extremal set. If $n$ is odd and $r=4$, then $g(n,r)=g([n],r)$, and $[n]$ is the unique rainbow $r$-extremal set. If $n$ is odd and $r\geq5$, then

$$g(n,r)=r^{\lceil n/2\rceil}\left(3-\frac{2}{r}\right),$$

and $I_3$ is the unique rainbow $r$-extremal set.

The paper proves the corresponding extremal results for $r\leq3$ and $r\geq8$; the exact structure remains unresolved for $4\leq r\leq7$, making these cases the central open part of the conjecture.