Conjecture on the asymptotic sharp threshold for Property (T) in random groups

Let $d_{(T)}(k)$ denote the sharp threshold for Property (T) in the $k$-gonal model.

Asymptotic threshold conjecture. The limit

$$\lim_{k \to \infty} d_{(T)}(k)$$

exists and is equal to the sharp threshold of the Gromov model.

This conjecture predicts that the behavior of $k$-gonal random groups approaches that of Gromov random groups as the polygonal length tends to infinity. The paper presents its results as evidence for the conjecture; its resolution is not stated here.