Logarithmic odd Dyson constant-term conjecture

Let $k$ and $m$ be positive integers. For variables $x_1,\ldots,x_{2k+1}$, let $\operatorname{CT}_{x_1,\ldots,x_{2k+1}}$ denote the constant term in their Laurent expansion. Then the logarithmic odd Dyson conjecture. Up to a sign,

$$\operatorname{CT}_{x_1,\ldots,x_{2k+1}} \frac{1}{(x_1 \cdots x_{2k+1})^{(2m+1)k}} \prod_{i=1}^k \operatorname{ln}\left(1-\frac{x_{2i}}{x_{2i-1}}\right) \prod_{1 \leq i <j \leq 2k+1 }(x_i-x_j)^{2m+1}$$

equals

$$\frac{((2k+1)(2m+1))!!}{(2k+1)!!(2m+1)!!^{2k+1}}.$$

This extends the preceding three-variable logarithmic constant-term identity in the direction of Dyson-type identities; the paper presents it as a precise conjecture, with no general proof supplied.