Kim's symmetry conjecture for crossing and nesting numbers of set partitions

Let $[n]=\{1,2,\ldots,n\}$, and let $P$ be a set partition of $[n]$. Its front representation is the linear representation obtained by joining consecutive elements in each block. For positive integers $i$ and $j$, let $\mathrm{cr}(P)$ and $\mathrm{ne}(P)$ denote its crossing and nesting numbers, respectively. Kim's conjecture. For any positive integers $i$ and $j$, the number of front representations of set partitions $P$ of $[n]$ with $\mathrm{cr}(P)=i$ and $\mathrm{ne}(P)=j$ equals the number of front representations of set partitions $P$ of $[n]$ with $\mathrm{cr}(P)=j$ and $\mathrm{ne}(P)=i$. The conjecture asserts symmetry of the joint distribution of crossing and nesting numbers for set partitions; the source states that the authors prove it.