Front crossing–nesting symmetry conjecture for set partitions

Let $\Pi_n$ be the set of set partitions of $[n]$, and let $\operatorname{FCR}(\pi)$ and $\operatorname{FNE}(\pi)$ denote respectively the front crossing and front nesting statistics of $\pi$. Front crossing–nesting symmetry conjecture. For every $n$,

$$\sum_{\pi\in\Pi_n} x^{\operatorname{FCR}(\pi)} y^{\operatorname{FNE}(\pi)}= \sum_{\pi\in\Pi_n} x^{\operatorname{FNE}(\pi)} y^{\operatorname{FCR}(\pi)}.$$

The conjecture asserts that the joint distribution of the front crossing and front nesting statistics is symmetric; it had been checked up to $n=11$ in the source, with no resolution supplied there.