Conjecture on the total cost of Quick-Find-Weighted

Let $C^{QFW}_{n,m}$ denote the total cost of the Quick-Find-Weighted algorithm through its first $m$ mergers, and let $\build{\longrightarrow }{}_{\mathcal L_2}$ denote convergence in $L^2$. Quick-Find-Weighted cost conjecture.

$$\frac{C^{QFW}_{n,n-1}}{n \log n} \build{\longrightarrow }{}_{\mathcal L_2} \frac{1}{\pi}.$$

The conjecture predicts a precise asymptotic constant for the total cost of Quick-Find-Weighted. The analogous result for Quick-Find-Biased, with limiting constant $1/2$, is proved in the preceding theorem; the authors also expect a smaller variance for Quick-Find-Weighted but do not provide a proof.