What is the best currently proven bounds on prime gaps?

I did some digging around on the internet but I found tons of different equations on both lower and upper bounds for the largest possible prime gap g(n). I was wondering what are currently the best bounds that have been proven.

I believe the best unconditional lower bound on $G(X)$, the largest gap for primes less than $X,$ is the one by Ford, Green, Konyagin, Maynard and Tao, discussed at Tao's blog at
$$g(X) \gg \log X \frac{\log_2 X \log_4 X} {\log_3 X}$$ where $\log_2(\cdot)=\log \log((\cdot))$, etc.
And the best known upper bound is $$G(X)\ll \frac{X^{0.525}}{\log X}$$ for $X$ large enough, due to Baker, Harman and Pintz.