Andrews–Curtis random-walk conjecture for black box normal subgroups

Let $G$ be a group, let $N\lhd G$, and define the Andrews–Curtis graph $\Delta_k(G,N)$ by

$$\Delta_k(G,N)=\left\{(h_1,\ldots,h_k)\mid \langle h_1^G,\ldots,h_k^G\rangle=N\right\}.$$

Its edges are given by the Andrews–Curtis moves described in the source, including multiplication by other entries and their conjugates.

Andrews–Curtis black-box conjecture. A random walk on the Andrews–Curtis graph $\Delta_k(G,N)$ provides a `good' black box for $N$.

The claim concerns a proposed modification of the product replacement algorithm for constructing practical black box oracles for normal subgroups. The supplied text gives no precise definition of `good' or evidence resolving the conjecture, so its status remains open.