Infinitesimal embedding conjecture for a random profinite action of a free group

Let $(t_n)_{n\geq 0}$ be a rapidly growing sequence of natural numbers, let $F_d$ be the free group, and construct a nested sequence of finite-index subgroups $(\Lambda_n)$ by taking $\Lambda_0$ uniformly at random among subgroups of index $t_0$ and, for $n\geq 1$, taking $\Lambda_n$ uniformly at random among subgroups of index $t_n$ in $\Lambda_{n-1}$. Let $(X,\mu)$ be the inverse limit of the actions on $F_d/\Lambda_n$ with normalized counting measures. Infinitesimal embedding conjecture. The action $F_d\curvearrowright (X,\mu)$ is infinitesimally embedded in $F_d$. This is presented as a candidate example of a compact, profinite action that may be infinitesimally contained in the acting group; no proof or counterexample is given.