  We show that the specialized quantum D-module of the equivariant quantum
cohomology ring of the minimal resolution of an ADE singularity is isomorphic
to the D-module of graded traces on the minimal nilpotent orbit in the Lie
algebra of the same type. This generalizes a recent result of Shlykov [Hikita
conjecture for the minimal nilpotent orbit, to appear in Proc. AMS,
https://doi.org/10.1090/proc/15281] and hence verifies in this case the quantum
version of Hikita's conjecture, proposed by Kamnitzer, McBreen and Proudfoot
[The quantum Hikita conjecture, Advances in Mathematics 390 (2021) 107947]. We
also show analogous isomorphisms for singularities of BCFG type.
