Norm–reciprocal-supernorm relation for the size ensemble

Let $W_{\rm size}(n)$ be the size-ensemble norm statistic and let $\widehat{W}_{\rm size}(n)$ be the corresponding non-cumulative reciprocal supernorm statistic. Size-ensemble norm–supernorm conjecture. One has

$$W_{\rm size}(n)\,\widehat{W}_{\rm size}(n) \sim 1$$

as $n \to \infty$. The source says this is equivalent to its size-statistic asymptotic conjecture via Lehmer's theorem, but does not establish it.