Kellner's conjecture on the distribution of prime factors of $\mathbb{D}^+_n$

For a prime $p$, let $s_p(n)$ be the sum of the base-$p$ digits of $n$, let $\omega(n)$ count the distinct prime divisors of $n$, and define

$$\mathbb{D}^+_n=\prod_{\substack{p>\sqrt{n}\s_p(n)\geq p}}p\qquad(n\geq1).$$

Kellner's conjecture. The following statements hold: $\mathbb{D}^+_n>1$, equivalently $\omega(\mathbb{D}^+_n)>0$, for $n>192$, and there exists a constant $\kappa>1$ such that

$$\omega(\mathbb{D}^+_n)\sim\kappa\frac{\sqrt n}{\log n}\qquad\text{as }n\to\infty.$$

The first assertion with $\kappa=2$ was proved by Bordellès et al. for sufficiently large $n$, while the asymptotic assertion is not resolved here.