Inducibility constants for the unresolved 3-cube configurations

Determine the inducibility constants for the three non-equivalent configurations \(W_3=\{(0,0,0)\}\), \(W_4=\{(0,0,0),(1,0,0)\}\), and \(W_5=\{(0,0,0),(0,0,1),(0,1,0)\}\) in \(Q_3\), whose best known lower bounds are respectively \(1/2\), \(4/9\), and \(1/3\).

Background

The paper resolves five previously difficult configurations in $\{0,1\}^3$, denoted by $W_7,W_8,W_9,W_{10},W_{12}$. It then identifies the remaining three non-equivalent 3-dimensional configurations for which the inducibility constant was not known. The authors report that the numerical upper bound for $W_5$ is close to, but does not coincide with, its known lower bound, indicating that resolving these cases may require methods beyond the calculations used in the paper.