Pick-function conjecture for normalized multiple Gamma functions

Let $G_n$ be the multiple Gamma functions of order $n$, defined by $G_1(z)=\Gamma(z)$ and the recurrence

$$G_{n+1}(z+1)=G_{n+1}(z)G_n(z),\qquad G_n(1)=1,$$

for $z\in\mathbb{C}$ and $n\in\mathbb{N}$. For $z\in\mathbb{C}\setminus(-\infty,0]$, define

$$f_n(z)=\frac{\log G_n(z+1)}{z^n\operatorname{Log}z}.$$

Pick-function conjecture. For every $n\in\mathbb{N}$, $f_n$ is a Pick function with Stieltjes representation

$$\frac{\log G_n(z+1)}{z^n\operatorname{Log}z}=\frac{1}{n!}-\int_0^\infty\frac{\tilde d_n(-t)}{t+z}\,dt,$$

where

$$\tilde d_n(t)=\begin{cases}-\dfrac{(-1)^{n-1}N_n(|t|)\log|t|+\log|G_n(t+1)|}{t^n\bigl((\log|t|)^2+\pi^2\bigr)},&t<0,\\0,&t\geq0,\end{cases}$$

and, for $t>0$,

$$N_n(t)=\frac{[t]([t]+1)([t]+2)\cdots([t]+n-1)}{n!}.$$

The cases $n=1$ and $n=2$ are known: the corresponding functions are Pick functions and their Stieltjes representations have been established. The conjecture asks for the analogous result for all orders $n$; establishing it is described as challenging.