Let $\mathcal{K}=\mathbb{C}((t))$ be the field of formal Laurent series over $\mathbb{C}$, and by $\mathcal{O}=\mathbb{C}[[t]]$ the ring of formal power series over $\mathbb{C}$.

Given a semi-simple Lie group $G$, affine Grassmannian $Gr_G$ is defined by the coset space $G(\mathcal K)/G(\mathcal O)$ wheareas the affine flag variety is defined by $Fl_G=G(\mathcal K)/I$ where $I$ is the Iwahori subgroup which is the preimage of a Borel subgroup $B$ under the map $G(\mathcal O)\to G$.

Bezrukavnikov-Finkelberg-Mirković showed that the $G(\mathcal{O})$-equivariant $K$-theory of affine Grassmannian $Gr_G$ is the coordinate ring of the following phase space $$ \textrm{Spec}\; K^{G(\mathcal{O})}(Gr_G)=(T\times T^\vee)/W $$ where $T$ is the maximal torus and W is the Weyl group.

My question is as follows: if you replace affine Grassmannian $Gr_G$ by affine flag variety $Fl_G$, then is the corresponding space

$$
\textrm{Spec}\; K^{G(\mathcal{O})}(Fl_G)
$$
just the direct product with the contangent bundle $T^*(G/B)$ of the flag variety (or nil-cone $\mathcal N$)
$$
((T\times T^\vee)/W)\times (T^*(G/B))~~~?
$$
Or is $\textrm{Spec}\; K^{G(\mathcal{O})}(Fl_G)$ a non-trivial $T^*(G/B)$-bundle (or some another bundle) over $(T\times T^\vee)/W$?