The denominators of harmonic numbers (Revised)

The denominators $d_n$ of the harmonic number $1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}$ do not increase monotonically with~$n$. It is conjectured that $d_n=D_n=\mathrm{L}\mathrm{C}\mathrm{M}(1,2,\dots ,n)$ infinitely often. For an odd prime $p$, the set $\{n:p{d}_n\mathrm{\mid }{D}_n\}$ has a harmonic density. Moreover, for $2<p_1<p_2<\cdots <p_k$, with $\log p_1\mathrm{/}\log p_i$ ($1\le i\le k$) being linearly independent, there exists $n$ such that $p_1{p}_2\cdots p_k{d}_n\mathrm{\mid }{D}_n$.