  We relate Reiner, Tenner, and Yong's coincidental down-degree expectations
(CDE) property of posets to the minuscule doppelg\"{a}nger pairs studied by
Hamaker, Patrias, Pechenik, and Williams. Via this relation, we put forward a
series of conjectures which suggest that the minuscule doppelg\"{a}nger pairs
behave "as if" they had isomorphic comparability graphs, even though they do
not. We further explore the idea of minuscule doppelg\"{a}nger pairs pretending
to have isomorphic comparability graphs by considering the rowmotion operator
on order ideals. We conjecture that the members of a minuscule doppelg\"{a}nger
pair behave the same way under rowmotion, as they would if they had isomorphic
comparability graphs. Moreover, we conjecture that these pairs continue to
behave the same way under the piecewise-linear and birational liftings of
rowmotion introduced by Einstein and Propp. This conjecture motivates us to
study the homomesies (in the sense of Propp and Roby) exhibited by birational
rowmotion. We establish the birational analog of the antichain cardinality
homomesy for the major examples of posets known or conjectured to have finite
birational rowmotion order (namely: minuscule posets and root posets of
coincidental type).
