Geometric realization conjecture for the symplectic Verlinde algebra completion

Let $G$ be the compact, simply connected Lie group under consideration, let $E_\ell G$ be the finite-dimensional approximations to $EG$, let $Ad(E_\ell G)$ denote the associated adjoint bundles, and let $K_*^\tau$ be the twisted $K$-homology theory with twist $\tau$. Write $V(\tau-h,G)$ for the corresponding Verlinde algebra and $I$ for its augmentation ideal. There is an increasing function

$$N:\mathbb{N}\to\mathbb{N}$$

and a collection of isomorphisms

$$f_\ell:K_*^\tau(Ad(E_\ell G))\cong V(\tau-h,G)/I^{N(\ell)}$$

which are coherent across the inverse system and induce the isomorphism of the completion theorem upon passage to the inverse limit.

Geometric realization conjecture. There is an increasing function $N:\mathbb{N}\to\mathbb{N}$ and a collection of isomorphisms

$$f_\ell:K_*^\tau(Ad(E_\ell G))\cong V(\tau-h,G)/I^{N(\ell)}$$

which are coherent across the inverse system—that is, they induce the isomorphism of the completion theorem upon passage to the limit.

The conjecture asks for a geometric, finite-stage realization of the inverse-limit isomorphism identifying twisted string $K$-homology with the completed Verlinde algebra. The supplied passage does not state whether this conjecture has been proved or refuted.