Lalonde–? higher homotopical stabilization conjecture for symplectic ball embeddings

Let $X$ be a symplectic manifold, and let $Emb_k(c_i)$ denote the space of embeddings of $k$ symplectic balls with capacities $c_i$. Let $\operatorname{Conf}_k(X)$ be the configuration space of $k$ points in $X$, and consider the forgetful map to the centers

$$\sigma:Emb_k(c_i)\to \operatorname{Conf}_k(X).$$

Higher homotopical stabilization conjecture. For every $m\in\mathbb{Z}^+$, there exists an $\epsilon(m)>0$ such that, whenever $c_i<\frac{\epsilon(m)}{k}$, the map $\sigma$ is $m$-connected.

This conjecture proposes a higher homotopical analogue of Biran's stabilization theorem, asserting that sufficiently small ball capacities make the space of symplectic ball embeddings approximate the configuration space through homotopy degree $m$.