Andrews' conjecture on partitions with no short sequences

Let $0<s<1$ and let $C_1,C_2,\ldots$ be independent events with probabilities

$${\bf P}_s(C_n)=1-e^{-ns}.$$

Let $A_k$ be the event

$$A_k=\bigcap_{i=1}^{\infty}(C_i\cup C_{i+1}\cup\cdots\cup C_{i+k-1}),$$

so that no sequence of $k$ consecutive events $C_i$ fails to occur. Andrews' conjecture. For each $k\geq 2$, there exists a positive constant $C_k$ such that

$${\bf P}_s(A_k)\sim C_k s^{-\frac12}\exp\left(-\frac{\lambda_k}{s}\right)\quad\text{as }s\downarrow0,$$

where

$$\lambda_k=\frac{\pi^2}{3k(k+1)}.$$

Andrews proposed this asymptotic using $q$-series identities. The paper proves it, so the conjecture is solved.