 Solution

At $\alpha =0$, the ordinary saddles coalesce at zero and $\sinh t-t=t^3/6+O(t^5)$. The contributing scale is $t=O({\nu }^{-1/3})$. The standard cubic saddle-point approximation givesEquivalently, the coefficient is $2^{1/3}\mathrm{Ai}(0)$.