The two-random-elements conjecture for full-dimensional closures

Let $\Gamma(p)$ be the profinite automorphism group of the rooted $p$-ary tree, and let the closure of a subgroup generated by random elements be considered in the profinite topology. Theorem 3random states that the subgroup of $\Gamma(p)$ generated by three random elements has $1$-dimensional closure with probability $1$. The two-random-elements conjecture. The same conclusion holds for two random elements: the subgroup of $\Gamma(p)$ generated by two random elements has $1$-dimensional closure with probability $1$. The preceding theorem establishes the analogous assertion for three random elements, while the source proposes the two-generator version without giving a resolution.