Nilsoliton stability conjecture for Ricci flow near nilpotent homogeneous metrics

Let $(\mathcal{M}^{n},g(t))$ be a solution of Ricci flow such that $g(0)$ is sufficiently close, in a suitable norm, to a (locally) homogeneous metric on a quotient of a nilpotent Lie group. A nilsoliton is a Ricci soliton metric on a nilpotent Lie group. Nilsoliton stability conjecture. There exists a limit solution $(\mathcal{M}_{\infty}^{n},g_{\infty}(t),x_{\infty})$ which is a nilsoliton. This is proposed as a stability mechanism for Type-III behavior of nilpotent, and more generally solvable, pseudosolitons; the suitable norm and the precise asymptotic blow-down construction are not specified, and the conjecture remains open.