The integrality conjecture for iterated harmonic numbers

For each integer $j\geq 1$, let $h_j(n)$ denote the $j$th iterated harmonic number, as defined in the paper. In particular, $h_j(1)=1$. Integrality conjecture. For each integer $j\geq 2$, the only $j$th iterated harmonic number that is an integer is

$$h_j(1).$$

The paper proves the corresponding assertion for ordinary harmonic numbers ($j=1$) and gives numerical evidence for $j=2$ and $j=3$; the assertion remains conjectural for all $j\geq 2$.