Concerning a conjecture of Taketomi-Tamaru

We study the setting of 2-step nilpotent Lie groups in the particular case that its type (p,q) is not exceptional. We demonstrate that, generically, the orbits of $\mathbb R^{>0}\times Aut_0$ in $GL(n)/O(n)$ are congruent even when a Ricci soliton metric does exists. In doing so, we provide a counterexample to the local version of a conjecture of Taketomi-Tamaru.