Rainbow-partition conjecture for meandric partners

Let $n$ be a positive integer. A meandric partner of a partition $\pi\in\mathrm{NC}(n)$ is a partition $\rho\in\mathrm{NC}(n)$ such that the associated meandric system $M(\pi,\rho)$ is a meander, equivalently such that $d_H(\pi,\rho)=n-1$. Let $r_n$ be the rainbow partition of $\{1,\ldots,n\}$, whose blocks are $\{1,n\},\{2,n-1\},\ldots$, together with a possible singleton block at $(n+1)/2$ when $n$ is odd. Let $K$ denote the Kreweras complementation map, and consider the orbit $\{K^j(r_n):0\leq j<n\}$. Rainbow-partition conjecture. The orbit of $r_n$ under Kreweras complementation is exactly the set of partitions in $\mathrm{NC}(n)$ having the largest number of meandric partners. Numerical experiments support this conjecture, which was also motivated by an idea raised in earlier work. By symmetry, the $n$ partitions in the orbit have the same number of meandric partners; determining whether there are any others with the same maximum remains open.