Kellner's large-prime-factor conjecture for Bernoulli-polynomial denominators

For a positive integer $n$, let $s_p(n)$ denote the sum of the base-$p$ digits of $n$, and define

$$\mathfrak P_n=\prod_{p\text{ prime}:\,s_p(n)\ge p}p.$$

Let $P(m)$ be the largest prime factor of $m$. Kellner's conjecture. For $n>192$,

$$P(\mathfrak P_n)>\sqrt n.$$

Equivalently, using $\mathfrak P_n^+=\prod_{p>\sqrt n,\,s_p(n)\ge p}p$, one has $\mathfrak P_n^+>1$ for every $n>192$. The source states that this is established, apart from the numerical threshold $192$, in a stronger form in its Theorem 2.