The logarithmic mean conjecture for the multivariate osculating mean

Let $n\geq 2$ and let $a_1,\ldots,a_n$ be positive, pairwise distinct real numbers. For the functions $x_1(t)=t,x_2(t)=t^2,\ldots,x_{n-1}(t)=t^{n-1},x_n(t)=\log t$, let $M_n(a_1,\ldots,a_n)$ denote the associated mean, and let $I_Z(a_1,\ldots,a_n)$ be the multivariate Identric mean defined by

$$I_Z(a_1,\ldots,a_n)=\exp\left[\frac{1}{V(a)}\sum_{i=1}^n(-1)^{n+i}a_i^{n-1}V_i(a)\log a_i-m\right].$$

Here $V(a)=\prod_{1\leq j<i\leq n}(a_i-a_j)$, $V_i(a)$ is the Vandermonde determinant obtained by deleting $a_i$, and $m=\sum_{k=1}^{n-1}\frac{1}{k}$.

Logarithmic mean conjecture. If $x_1(t)=t$, $x_2(t)=t^2$, $\ldots$, $x_{n-1}(t)=t^{n-1}$, and $x_n(t)=\log t$, then

$$M_n(a_1,\ldots,a_n)=I_Z(a_1,\ldots,a_n).$$

The assertion extends the verified three-variable identity $M_3(a,b,c)=I_Z(a,b,c)$ and would identify the osculating mean associated with these functions with the multivariate Identric mean in every dimension. The source provides no resolution beyond this formulation.