$K$-theoretic quantum Hikita conjecture at roots of unity

Let $X$ and $X^!$ be symplectically dual varieties arising from Coulomb-branch and resolved Higgs-branch constructions for the gauge theory $(G,N)$. Let $\zeta$ be the relevant root of unity, and let $B_z(X)$ and $z^{\mathrm{eff}}$ denote the root-of-unity coefficient algebra and effective degree variables used in the source. $K$-theoretic quantum Hikita conjecture at roots of unity. There is an isomorphism of specialized $q$-difference modules

$$M_{q,eq}(X)|_{q=\zeta} \cong M_{q,Kah}(X^!)|_{q=\zeta}$$

that is equivariant with respect to a ring isomorphism

$$B_z(X) \cong K_{\mathbb{G}_m}(X^!)[[z^{\mathrm{eff}}]]$$

and the actions of Frobenius-constant quantizations and quantum Adams operators. This is an arithmetic refinement of the $K$-theoretic quantum Hikita conjecture, proposed as a conjectural compatibility at roots of unity.