Kellner's asymptotic conjecture for large prime factors of Bernoulli-polynomial denominators

For a positive integer $n$, define

$$\mathfrak P_n^+=\prod_{p>\sqrt n,\,s_p(n)\ge p}p,$$

where $s_p(n)$ is the sum of the base-$p$ digits of $n$. Let $\omega(m)$ denote the number of distinct prime factors of $m$. Kellner's conjecture. There is an absolute constant $\kappa>0$ such that, as $n\to\infty$,

$$\omega(\mathfrak P_n^+)=(\kappa+o(1))\frac{\sqrt n}{\log n}.$$

This predicts the precise order of growth of the number of large prime factors occurring in the denominator of the Bernoulli polynomial. The source gives no resolution of this asymptotic conjecture.