Conjecture on the limiting distribution of random Cayley graph diameters

Let $q$ be prime, let $g_1,\ldots,g_k$ be independent uniformly random generators of the cyclic group ${\mathbb Z}_q$, and let $Diam(q,k)$ denote the diameter of the resulting directed Cayley graph. The same notation may be used for the corresponding undirected Cayley graph with generating set $\{g_1,-g_1,\ldots,g_k,-g_k\}$. Diameter distribution conjecture. The normalized diameter

$$\frac{Diam(q,k)}{\sqrt[k]{q}}$$

converges in distribution to some distribution $D(k)$ on ${\mathbb R}$ whose support is non-compact. The preceding upper and lower bounds show that the normalization by $\sqrt[k]{q}$ is of the correct order and that the limiting behavior is non-degenerate in the stated sense; the existence and non-compactness of the limiting distribution remain open in the source.