Asymptotic inversion constant for the non-reduced random permutation

Let $p\in[0,1)$ and let $\boldsymbol{w}=\boldsymbol{w}(n)\in S_n$ be the random permutation in the non-reduced model. Write $\operatorname{inv}(\boldsymbol{w})$ for its number of inversions. Inversion limit conjecture. There is a constant $\varkappa$ such that

$$\frac{\operatorname{inv}(\boldsymbol{w})}{n^{3/2}}\longrightarrow \varkappa\sqrt{\frac{p}{1-p}}$$

in probability as $n\to\infty$. The preceding bounds imply only that $2/(3\sqrt{\pi})\leq\varkappa\leq4/(3\sqrt{\pi})$; the source gives no resolution of this conjecture.