The rainbow Sidon-coloring conjecture

Let $r\ge 2$ be an integer and let $n$ be sufficiently large. An $r$-coloring of a subset of $[n]$ is rainbow $[1,-1,1,-1]$-free if it contains no rainbow $4$-element set satisfying the corresponding linear equation. The rainbow Sidon-coloring conjecture. Among all subsets of $[n]$, the full set $[n]$ is the unique subset admitting the maximum number of rainbow $[1,-1,1,-1]$-free $r$-colorings. In particular, almost all rainbow $[1,-1,1,-1]$-free $r$-colorings of $[n]$ use at most $3$ colors. This is motivated by the connection between $[1,-1,1,-1]$-free sets and Sidon sets, and predicts both extremality of the full interval and a three-color concentration phenomenon for almost all such colorings.