Push-forward of geometric distributions under Collatz iteration: Part 1

Two conjectures are presented. The first, Conjecture 1, is that the pushforward of a geometric distribution on the integers under $n$ Collatz iterates, modulo $2^p$, is usefully close to uniform distribution on the integers modulo $2^p$, if $p\mathrm{/}n$ is small enough. Conjecture 2 is that the density is bounded from zero for the incidence of both $0$ and $1$ for the coefficients in the dyadic expansions of $-3^{-\mathrm{\ell }}$ on all but an exponentially small set of paths of a geometrically distributed random walk on the two-dimensional array of these coefficients. It is shown that Conjecture 2 implies Conjecture 1. At present, Conjecture 2 is unresolved.