Universal limiting profile for successive minimum spanning tree forests

For each $k\geqslant1$, let $F_k(t)$ be the forest produced by Kruskal's algorithm at time $t$, and let $\rho_k(t)$ be the limiting fraction of vertices in its largest component. The functions $\rho_k$ describe the giant-component sizes of the successive forests. Universal profile conjecture. There exists a continuous increasing function $\rho_\infty:(-\infty,\infty)\to[0,1)$ such that

$$\rho_k(2k+x)\to\rho_\infty(x)$$

uniformly in $x\in\mathbb R$ as $k\to\infty$. This predicts that the giant-component profiles become translates of one universal profile; the paper reports computational evidence, but no proof.