  Recently, Panyushev raised five conjectures concerning the structure of
certain root posets arising from $\mathbb{Z}$-gradings of simple Lie algebras.
This paper aims to provide proofs for four of them. Our study also links these
posets with Kostant-Macdonald identity, minuscule representations, Stembridge's
"$t=-1$ phenomenon", and the cyclic sieving phenomenon due to Reiner, Stanton
and White.
