Density of Unbounded Aliquot Sequences

Question

Let $s_1(n)=\big{(}\sum_{d|n}d\big{)}-n=\sigma_1(n)-n$ be the restricted divisor sum, and define $s_k(n)=s_1(s_{k-1}(n))$ as the $k^{th}$ term of the aliquot sequence starting at $n$. The Catalan Dickson Conjecture states that every sequence is bounded; in other words, every sequences either terminates at 1 or enters a cycle. Though this conjecture is not proven to date, I was wondering if any weaker forms of the conjecture have been proven. For example, has any statement been made regarding the density of integers which have non-terminating sequence (clearly a nonzero density would imply infinitude, but has it been shown that they might have zero density)? Guy and Selfridge argue that most even sequences should diverge, and most odd sequences should terminate, so would it be reasonable to conjecture that the density of non-terminating aliquot sequences approaches a half?