Very good question! Yes, the final value is somewhere in there. I expect that it's the lowest point, meaning Croft's Tortoise is optimal. Since then we have an unpublished new 0.241 upper bound with the same proof but wider computer search.
We don't know what Erdős had in mind when picking 1/4, but we know something that seems to make the 1/4 value special. Many of the earlier attempts to prove Erdős's conjecture were based on a notion called fractional chromatic number. The numbers went down like this: 0.2857, 0.2813, 0.2763, 0.2565, 0.2518, 0.2506. We now have a new preprint that reaches exactly 1/4, not more not less, and we don't know how to improve it: https://arxiv.org/abs/2311.10069
So it seems like the fractional chromatic number based approach has an inherent barrier at 1/4. If that is true, those earlier fractional chromatic number based attempts were doomed, and we only managed to break that barrier because we used some extra ideas. (Namely, Fourier analysis).
We don't know what Erdős had in mind when picking 1/4, but we know something that seems to make the 1/4 value special. Many of the earlier attempts to prove Erdős's conjecture were based on a notion called fractional chromatic number. The numbers went down like this: 0.2857, 0.2813, 0.2763, 0.2565, 0.2518, 0.2506. We now have a new preprint that reaches exactly 1/4, not more not less, and we don't know how to improve it: https://arxiv.org/abs/2311.10069
So it seems like the fractional chromatic number based approach has an inherent barrier at 1/4. If that is true, those earlier fractional chromatic number based attempts were doomed, and we only managed to break that barrier because we used some extra ideas. (Namely, Fourier analysis).