  The $k$-gonal models of random groups are defined as the quotients of free
groups on $n$ generators by cyclically reduced words of length $k$. As $k$
tends to infinity, this model approaches the Gromov density model. In this
paper we show that for any fixed $d_0 \in (0, 1)$, if positive $k$-gonal random
groups satisfy Property (T) with overwhelming probability for densities $d
>d_0$, then so do $nk$-gonal random groups, for any $n \in \mathbb{N}$. In
particular, this shows that for densities above 1/3, groups in $3k$-gonal
models satisfy Property (T) with probability 1 as $n$ approaches infinity.
