A proof of Andrews' conjecture on Partitions with no short sequences

Holroyd, Liggett, and Romik introduced the following probability model. Let $C_1, C_2,...$ be independent events with probabilities $\P_s(C_n)= 1-e^{-ns}$ under a probability measure $\P_s$ with $0<s<1$. Let $A_k$ be the event that there is no sequence of $k$ consecutive $C_i$ that do not occur. We given an asymptotic for $\P_s(A_k)$ with a relative error term that goes to 0 as $s\to 0$. This establishes a conjecture of Andrews.