  For each minuscule flag variety $X$, there is a corresponding minuscule
poset, describing its Schubert decomposition. We study an action on plane
partitions over such posets, introduced by P. Cameron and D. Fon-der-Flaass
(1995). For plane partitions of height at most $2$, D. Rush and X. Shi (2013)
proved an instance of the cyclic sieving phenomenon, completely describing the
orbit structure of this action. They noted their result does not extend to
greater heights in general; however, when $X$ is one of the two minuscule flag
varieties of exceptional Lie type $E$, they conjectured explicit instances of
cyclic sieving for all heights.
  We prove their conjecture in the case that $X$ is the Cayley-Moufang plane of
type $E_6$. For the other exceptional minuscule flag variety, the Freudenthal
variety of type $E_7$, we establish their conjecture for heights at most $4$,
but show that it fails generally. We further give a new proof of an unpublished
cyclic sieving of D. Rush and X. Shi (2011) for plane partitions of any height
in the case $X$ is an even-dimensional quadric hypersurface. Our argument uses
ideas of K. Dilks, O. Pechenik, and J. Striker (2017) to relate the action on
plane partitions to combinatorics derived from $K$-theoretic Schubert calculus.
