Strong-extension exact theorem without the hermetic assumption

Prove an exact extremal-graph theorem for forbidden-colouring families satisfying the strong extension property, replacing the stronger hypotheses that the family has both the extension property and is hermetic.

Background

The paper's exact theorem requires boundedness, hermeticity, and the extension property. The authors note that hermeticity is a strong condition and conjecture that the proof strategy should extend under the weaker strong extension property alone. Such a result would cover additional non-monochromatic patterns whose optimization problems are tractable.

References

We see no obstacle to proving an analogue of the exact result in which would replace the assumption that $X$ has the extension property and is hermetic with the assumption that $X$ has the strong extension property.

— 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 6.2, 'A more general exact result'