Approximation conjecture for steady states in the sigmoidal bounded-confidence model

Let $\mathbf{x}$ be a steady state of a Hegselmann–Krause model with confidence bound $\sqrt{\delta}$. Define $C_\delta(\mathbf{x})\subset\mathbb{R}^n$ to be the set of opinion vectors $\mathbf{y}$ satisfying

$$(y_i-y_j)^2\leq\delta\iff (x_i-x_j)^2\leq\delta\quad\text{for all }i\sim j.$$

Approximation conjecture. There exist $\mathbf{x}'\in C_\delta(\mathbf{x})$, a sequence $\{\gamma^{(\ell)}\}_\ell$, and a sequence $\{\mathbf{x}^{(\ell)}\}_\ell$ such that

$$\mathbf{F}_{\gamma^{(\ell)}}(\mathbf{x}^{(\ell)})=\mathbf{0}$$

and $\mathbf{x}^{(\ell)}\to\mathbf{x}'$ as $\gamma\to\infty$. The conjecture asserts that every pattern of mutual influence represented by an HK steady state has a representative opinion vector approximable by steady states of the SBCM.