Orbits of plane partitions of exceptional Lie type

We prove their conjecture in the case that X is the Cayley-Moufang plane of type E6. For the other exceptional minuscule flag variety, the Freudenthal variety of type E7, 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 We further give a new proof of an unpublished cyclic sieving of Rush and Shi (2011) for plane partitions of any height in the case X is an even-dimensional quadric hypersurface. Our argument uses ideas of Dilks et al. (2017) to relate the action on plane partitions to combinatorics derived from K-theoretic Schubert calculus.