The quantum Hikita conjecture for symplectically dual resolutions

Let $X$ and $X^!$ be symplectically dual conical Hamiltonian symplectic resolutions, with equivariant and Kähler roots, rings of differential operators, and associated $D$-modules as defined for the pair. Quantum Hikita conjecture. There exist a bijection of roots, a compatible isomorphism of rings of differential operators, and a compatible isomorphism of regular $D$-modules taking $1$ to $1$, extending the Hikita--Nakajima isomorphisms. For every $a$ in the relevant $B$-algebra, this isomorphism intertwines the Frobenius-constant quantization action on the trace-side module with the specialized quantum Steenrod action on the quantum-side module. This conjecture extends the Hikita--Nakajima conjecture by incorporating root data and $D$-modules; the paper proposes its mod-$p$ refinement and verifies the corresponding statements in Springer and hypertoric examples.