The conjecture that the maximum score of a decomposition-star space is $8\,\mathrm{pt}$

The article defines a compact topological space $X$ of decomposition stars and a continuous function $\sigma$ on $X$. Let

$$\mathrm{pt}=-\frac{\pi}{3}+\sqrt{2}\,\delta_{\mathrm{tet}},$$

where $\delta_{\mathrm{tet}}$ is the packing density associated with a regular tetrahedron of edge length $2$; thus $\mathrm{pt}\approx 0.05537$. The constant $8\,\mathrm{pt}$ is approximately $0.442989$.

Maximum-score conjecture. The maximum of $\sigma$ on $X$ is the constant

$$8\,\mathrm{pt}\approx 0.442989.$$

This conjecture is presented as part of the finite-dimensional optimization approach to the Kepler conjecture. The supplied text gives no resolution status for this auxiliary conjecture.