The denominators of harmonic numbers (Revised)

The denominators $d_n$ of the harmonic number $1+\frac12+\frac13+\cdots+\frac1n$ do not increase monotonically with~$n$. It is conjectured that $d_n=D_n={\rm LCM}(1,2,\ldots,n)$ infinitely often. For an odd prime $p$, the set $\{n:pd_n|D_n\}$ has a harmonic density. Moreover, for $2<p_1<p_2<\cdots<p_k$, with $\log p_1/\log p_i$ ($1\le i\le k$) being linearly independent, there exists $n$ such that $p_1p_2\cdots p_kd_n|D_n$.