Reciprocal supernorm max-part-statistic asymptotic conjecture

Let ${\rm Max}(n)$ be the ensemble of partitions with largest part $n$, and let $\widehat{W}_{\rm \max}(n)$ denote its non-cumulative reciprocal supernorm statistic. Max-part-ensemble conjecture. One has

$$\widehat{W}_{\rm \max}(n) \sim \frac{e^{\gamma}}{n}$$

as $n \to \infty$. This is the analogous prediction for individual max-part ensembles; the source gives no data because finite values require infinite summations, and no proof or resolution is stated.