Born winner/silver spoon regime conjecture for the multiplicative coalescent

Let $(\boldsymbol{x}^{\scriptscriptstyle(n)})_{n\geqslant 1}$ satisfy Condition II$\tau$ for some $\tau\in(3,4)$, and let $M_n$ be as in the leader theorem. Write $\mathcal{W}_n$ for the event that the component containing $x_1^{\scriptscriptstyle(n)}$ is always the leader, meaning that the identity of the leader never changes. Assume $c_1>c_2$. Born winner/silver spoon regime conjecture. There exists $\delta>0$ such that

$$\liminf_{n\to\infty}\operatorname{\mathbb{P}}(M_n^-\geqslant K)>\delta$$

for every $K>0$. Moreover,

$$\liminf_{n\to\infty}\operatorname{\mathbb{P}}(\mathcal{W}_n)>0.$$

The first assertion predicts that, under Condition II$\tau$, the negative part of the leader-stabilization time is not tight, complementing the tightness result for its positive part. The second asserts a positive limiting probability that the component containing the largest initial particle remains leader forever; both claims are left conjectural in the stated regime.