Largest-prime-gap asymptotic via the interval sieve

Let $W_y$ be the minimum of $|[0,y]\cap\mathcal{S}_{(y/\log y)^{1/2}}|$ over all choices of residue classes modulo primes up to $(y/\log y)^{1/2}$, and define

$$g(u)=\max\{y:W_y\log y\le u\}.$$

Let $G_{\mathcal{P}}(x)$ denote the largest gap between consecutive primes up to $x$, and let $\xi=2e^{-\gamma}$. Asymptotic for largest gap in the primes.

$$g((\xi-o(1))\log^2x)\lesssim G_{\mathcal{P}}(x)\lesssim g((\xi+o(1))\log^2x)\qquad(x\to\infty).$$

This is the prediction obtained by transferring the rigorously analyzed random-sieve model to the primes. The stronger prediction $G_{\mathcal{P}}(x)\sim\xi\log^2x$ is conditional on an additional folklore conjecture about the interval sieve.