The constant-term conjecture for W-algebra extensions of (2,p) minimal models

Let $k$ and $m$ be positive integers. For a Laurent series in $x_1,\ldots,x_{2k+1}$, write ${\rm CT}_{x_1,\ldots,x_{2k+1}}$ for its constant term, and let $r!!$ denote the double factorial. Then, up to a sign,

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

Constant-term conjecture. This constant term equals

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

The conjecture would determine the nonzero constant left unspecified in the preceding corollary and is presented as an analogue of a conjecture from the cited earlier work; its status is not resolved in the supplied text.