Reciprocal supernorm perimeter-statistic asymptotic conjecture

Let ${\rm Per}(n)$ be the perimeter ensemble of partitions indexed by $n$, and let $\widehat{W}_{\rm per}(n)$ denote its non-cumulative reciprocal supernorm statistic. Perimeter-ensemble conjecture. One has

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

as $n \to \infty$. This conjecture predicts the differentiated form of the cumulative perimeter-ensemble asymptotic, with Euler's constant as the leading constant; the source presents numerical comparison but no proof or resolution.