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I pulled one of the tables of prime gaps off wikipedia and put a final column, $g/\log^2p,$ just as in the section in Guy's book. For $11 \leq p < 4 \cdot 10^{18},$ we have $g < \log^2p.$ Completely unprovable for larger $p.$ After line 3 (prime is 7) the closest we get to $1$ is line 64, $ \; g = 1132, \; p \approx 1.69 \cdot 10^{15}, \; g/\log^2p \approx 0.920639.$ I believe Cramer-Granville is the conjecture that $ \; \limsup g/\log^2 p$ is finite, and the disagreement is over whether it is more likely to be $1$ or something else. However, it gives an opinion on your original question.

I pulled one of the tables of prime gaps off wikipedia and put a final column, $g/\log^2p,$ just as in the section in Guy's book. For $11 \leq p < 4 \cdot 10^{18},$ we have $g < \log^2p.$ Completely unprovable for larger $p.$ After line 3 (prime is 7) the closest we get to $1$ is line 64, $ \; g = 1132, \; p \approx 1.69 \cdot 10^{15}, \; g/\log^2p \approx 0.920639.$ I believe Cramer-Granville is the conjecture that $ \; \limsup g/\log^2 p$ is nonzero but finite, and the disagreement is over whether it is more likely to be $1$ or something else. However, it gives an opinion on your original question.

I pulled one of the tables of prime gaps off wikipedia and put a final column, $g/\log^2p,$ just as in the section in Guy's book. For $11 \leq p < 4 \cdot 10^{18},$ we have $g < \log^2p.$ Completely unprovable for larger $p.$ After line 3 (prime is 7) the closest we get to $1$ is line 64, $ \; g = 1132, \; p \approx 1.69 \cdot 10^{15}, \; g/\log^2p \approx 0.920639$

I pulled one of the tables of prime gaps off wikipedia and put a final column, $g/\log^2p,$ just as in the section in Guy's book. For $11 \leq p < 4 \cdot 10^{18},$ we have $g < \log^2p.$ Completely unprovable for larger $p.$ After line 3 (prime is 7) the closest we get to $1$ is line 64, $ \; g = 1132, \; p \approx 1.69 \cdot 10^{15}, \; g/\log^2p \approx 0.920639.$ I believe Cramer-Granville is the conjecture that $ \; \limsup g/\log^2 p$ is finite, and the disagreement is over whether it is more likely to be $1$ or something else. However, it gives an opinion on your original question.

I pulled one of the tables of prime gaps off wikipedia and put a final column, $g/\log^2p,$ just as in the section in Guy's book. For $11 \leq p < 4 \cdot 10^{18},$ we have $g < \log^2p.$ Completely unprovable for larger $p.$

I pulled one of the tables of prime gaps off wikipedia and put a final column, $g/\log^2p,$ just as in the section in Guy's book. For $11 \leq p < 4 \cdot 10^{18},$ we have $g < \log^2p.$ Completely unprovable for larger $p.$ After line 3 (prime is 7) the closest we get to $1$ is line 64, $ \; g = 1132, \; p \approx 1.69 \cdot 10^{15}, \; g/\log^2p \approx 0.920639$