Extremality among almost optimal complete multipartite graphs for (3,5)

Determine which graphs in the large family of complete multipartite graphs that are asymptotically almost extremal for the monochromatic triangle problem with five colours are actually extremal.

Background

For the monochromatic triangle problem with five colours, previous work identified a large family of complete multipartite graphs whose numbers of valid colourings are asymptotically optimal. However, asymptotic optimality alone does not determine the exact extremal graph or graphs, leaving the classification unresolved.

References

However, for $(3,5)$ there is a large family of graphs which are almost extremal, though we do not know which of these are extremal; nevertheless, these graphs are all complete multipartite.

— A framework for the generalised Erdős-Rothschild problem and a resolution of the dichromatic triangle case  (2502.12291 - Gupta et al., 17 Feb 2025) in Section 1.1, 'The monochromatic Erdős–Rothschild problem'