Adamović–Milas logarithmic constant term identity

Let $(a)_m=a(a+1)\cdots(a+m-1)$ be the rising factorial, let $\operatorname{CT}$ denote the constant term, and define the Vandermonde product by

$$\Delta(X)=\prod_{1\leq i<j\leq n}(x_i-x_j).$$

Adamović–Milas logarithmic constant term conjecture. For odd positive integer $n$ and nonnegative integer $k$, set $m=(n-1)/2$ and $K=2k+1$. Then

$$\operatorname{CT}\bigg[\Delta(X) \prod_{i=1}^n x_i^{-m} \prod_{i=1}^m \log\Big(1-\frac{x_{2i}}{x_{2i-1}}\Big) \prod_{1\leq i\neq j\leq n}\Big(1-\frac{x_i}{x_j}\Big)^k\bigg]=\frac{(nK)!!}{n!!(K!!)^n}.$$

This is presented as a discovery of Adamović and Milas and as a logarithmic analogue of the equal-parameter Dyson identity. The supplied text gives no resolution status for this identity.