Asymptotic growth conjecture for graph-complex homotopy ranks

Let $\overline{\operatorname{Emb}}_c(\mathbb{R}^m,\mathbb{R}^n)$ denote the homotopy fibre of the inclusion of embeddings into immersions, and let $\pi_*^{(t-\ell,t)}$ denote its component of complexity $t$ and Hodge degree $t-\ell$. For fixed $\ell$, the asymptotic growth conjecture asserts that the rank of

$$\pi_*^{(t-\ell,t)}(\overline{\operatorname{Emb}}_c(\mathbb{R}^m,\mathbb{R}^n))$$

has asymptotic $t^\ell$ as $t\to +\infty$. The conjecture is motivated by computations for the subcomplex with two cycles, where the homology ranks grow linearly with $t$; the general asymptotic claim remains open.