Conjecture on convergence from free to Boolean maximum powers

Let $\{F_n\}_n$ be a sequence in $\Delta_+$ and $\{k_n\}_n$ a sequence of positive integers such that

$$k_1<k_2<\cdots.$$

Suppose that $F\in\Delta_+$ and that

$$F_n^{\Box\hspace{-.55em}\lor k_n}\xrightarrow{w}F.$$

Free-to-Boolean maximum convergence conjecture. Then there exists $G\in\Delta_+$ such that

$$F_n^{\cup\hspace{-.52em}\lor k_n}\xrightarrow{w}G\qquad\text{as }n\to\infty.$$

Moreover,

$$G=\frac{1}{2-F}$$

on $\{F>0\}$. This conjecture formulates the expectation that the converse implication from free maximum convergence to Boolean maximum convergence holds without assuming that the limiting distribution is positive everywhere. The examples preceding the conjecture show that the behavior on $\{F=0\}$ can depend on the approximating sequence, while the claimed existence of a distribution-function limit and the formula on $\{F>0\}$ remain to be established.