The straightbrush model conjecture for the residual compact set

Let $K_{\mathrm{addr}}$ be the addresses odometer, identified with the Cantor set of connected components of $K_\infty$. A subset $B\subset K_{\mathrm{addr}}\times[0,1]$ is required to encode each component as a vertical initial segment. Straightbrush model conjecture. There exists a closed subset $B$ of $K_{\mathrm{addr}} \times [0,1]$ homeomorphic to $K_\infty$ which satisfies

1. $K_{\mathrm{addr}}\times\{0\}\subset B$;
2. For all $\alpha\in K_{\mathrm{addr}}$ there exists $e_\alpha\in[0,1]$ such that $(\alpha,t)\in B$ if and only if $0\leq t\leq e_\alpha$ (such $e_\alpha$ is called an upper endpoint);
3. The set of upper endpoints is dense in $B$.

This conjectures a topological model of $K_\infty$ analogous to a straightbrush or hairy arc. The source establishes a dense countable family of components homeomorphic to a common line segment, but leaves the global homeomorphism to this model open.