In "The root poset and its relatives" (https://arxiv.org/abs/math/0502385) Panyushev established (see Corollary 3.4) that the average size of an antichain of the root poset $\Phi^+$ of an irreducible crystallographic root system $\Phi$ is $n/2$, where $n$ is the number of simple roots of $\Phi$. He did this just by observing that the "$\Phi$-Narayana polynomial", i.e., the generating function for antichains of $\Phi^+$ by cardinality, is palindromic, something he in fact observed in his earlier paper "ad-nilpotent ideals of a Borel subalgebra: generators and duality" (https://www.sciencedirect.com/science/article/pii/S0021869303006380) (see Section 6).
In the "ad-nilpotent ideals" paper he is very interested in the fact that the Narayana polynomials are palindromic and uses this palindromicness as evidence in favor of a certain conjectured "duality" for the ad-nilpotent ideals (which correspond bijectively to antichains of the root post). In particular he conjectures (see Conjecture 6.1) the existence of a natural involution on the set of antichains which would send an antichain of cardinality $k$ to an antichain of cardinality $n-k$. As far as I know, this conjecture remains open (see my MO question Panyushev's conjectured duality for root poset antichains).
He couldn't prove the existence of that duality, but in the "On orbits of antichains of positive roots" paper (https://arxiv.org/abs/0711.3353) he accomplished something close using "rowmotion"/"the Fon-der-Flaass operator". Namely, he conjectured a specific way to partition the set of antichains of $\Phi^+$ into "small" sets (of size dividing $2h$ where $h$ is the Coxeter number of $\Phi$) such that in each such set the average size is $n/2$. (So the conjectured involution on the set of antichains would be doing the same thing except with sets of size dividing $2$ instead of $2h$.)
In this way I view Panyushev's homomesy conjecture as an extension of his investigation of the "duality" property for ad-nilpotent ideals, which evolved over the course of the three papers referenced above.
EDIT: Here is some further "context"/speculation:
Let $\Phi$ an irreducible crystallographic root system with Weyl group $W$. As mentioned above, the $\Phi$-Narayana number $N_k(\Phi)$ is the number of antichains of the root poset $\Phi^+$ of cardinality $k$. And the $\Phi$-Narayana polynomial is then $N(\Phi;q) = \sum_{k=0}^{n} N_k(\Phi)q^k$. Note that $N(\Phi;1)=\mathrm{Cat}(\Phi)$ is the $\Phi$-Catalan number.
The previous paragraph gave a "nonnesting" description of $N(\Phi;q)$. There is also a "noncrossing" description. Namely, recall that the lattice of noncrossing partitions of $\Phi$, denoted $NC(\Phi)$, is the induced subposet of the absolute order on $W$ below $c$, where $c$ is any fixed Coxeter element of $W$. Then $NC(\Phi)$ is a graded poset, and the Narayana number $N_k(\Phi)$ is also the number of elements of $NC(\Phi)$ at rank $k$ (so $N(\Phi;q)$ is the rank generating function for $NC(\Phi)$). For these (and other) descriptions of $N(\Phi;q)$, see Theorem 5.9 of Fomin-Reading "Root systems and generalized associahedra" (https://arxiv.org/abs/math/0505518).
Now, as mentioned, Panyushev was very interested in the fact that $N(\Phi;q)$ is palindromic, which is not at all obvious from the nonnesting description. In particular he was looking for a bijective (in fact, involutive) proof of this fact. But there is a nice way to see that $N(\Phi;q)$ is palindromic from the noncrossing description. Namely, there is the Kreweras complementation map $\mathrm{Krew}\colon w \mapsto cw^{-1}$ on $NC(\Phi)$, which takes an element of rank $k$ to an element of rank $n-k$.
Note that while $\mathrm{Krew}$ does provide a bijective proof that $N(\Phi;q)$ is palindromic, $\mathrm{Krew}$ is not an involution (it has order $h$ or $2h$). In fact, here's where the connection to Panyushev's "rowmotion" action comes in: elements of $NC(\Phi)$ under $\mathrm{Krew}$ are in equivariant bijection with antichains of $\Phi^+$ under rowmotion. This was conjectured by Bessis-Reiner (https://arxiv.org/abs/math/0701792) and proved by Armstrong-Stump-Thomas (https://arxiv.org/abs/1101.1277).
As far as I can tell, Panyushev did not know at the time that his action was the "same" as Kreweras complementation. But I do think it's interesting that he came up with his action with the goal of understanding why $N(\Phi;q)$ is palindromic on the nonnesting side (or at least why the average antichain cardinality is $n/2$), while the Kreweras complementation proves this on the noncrossing side.